Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 92.2%

Time: 32.3s

Alternatives: 17

Speedup: 3.8×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% 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: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -230000 \lor \neg \left(t \leq 6.4 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -230000.0) (not (<= t 6.4e-93)))
   (/
    (pow
     (* (/ (cbrt (/ 2.0 (tan k))) t) (* (cbrt l) (/ (cbrt l) (cbrt (sin k)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (cos k) t)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -230000.0) || !(t <= 6.4e-93)) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * (cbrt(l) * (cbrt(l) / cbrt(sin(k))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -230000.0) || !(t <= 6.4e-93)) {
		tmp = Math.pow(((Math.cbrt((2.0 / Math.tan(k))) / t) * (Math.cbrt(l) * (Math.cbrt(l) / Math.cbrt(Math.sin(k))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -230000.0) || !(t <= 6.4e-93))
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * Float64(cbrt(l) * Float64(cbrt(l) / cbrt(sin(k))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -230000.0], N[Not[LessEqual[t, 6.4e-93]], $MachinePrecision]], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.3e5 or 6.3999999999999997e-93 < 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* 59.2%

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

      3. sqr-neg 59.2%

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

      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-/r* 70.2%

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

   3. Simplified 70.7%

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

      1. associate-/l/ 70.7%

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

      2. associate-/r/ 70.2%

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

      3. add-cube-cbrt 70.1%

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

   5. Applied egg-rr 78.3%

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

      1. pow-plus 78.3%

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

      2. metadata-eval 78.3%

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

      3. associate-/r/ 78.4%

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

      4. associate-/r/ 78.3%

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

   7. Simplified 78.3%

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

      1. cbrt-prod 91.4%

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

   9. Applied egg-rr 91.4%

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

       1. cbrt-div 95.0%

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

   11. Applied egg-rr 95.0%

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

   if -2.3e5 < t < 6.3999999999999997e-93

   1. Initial program 38.7%

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

      1. associate-/r* 38.7%

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

      2. associate-*l* 38.7%

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

      3. sqr-neg 38.7%

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

      4. associate-*l* 38.7%

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

      5. *-commutative 38.7%

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

      6. sqr-neg 38.7%

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

      7. associate-/r* 38.7%

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

   3. Simplified 38.7%

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

      1. associate-/l/ 38.7%

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

      2. associate-/r/ 38.7%

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

      3. add-cube-cbrt 38.7%

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

   5. Applied egg-rr 55.8%

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

      1. pow-plus 55.8%

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

      2. metadata-eval 55.8%

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

      3. associate-/r/ 55.8%

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

      4. associate-/r/ 55.8%

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

   7. Simplified 55.8%

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

      1. cbrt-prod 55.5%

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

   9. Applied egg-rr 55.5%

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

       1. cbrt-div 55.4%

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

   11. Applied egg-rr 55.4%

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

   12. Taylor expanded in k around inf 78.9%

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

       1. times-frac 78.8%

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

       2. unpow2 78.8%

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

       3. associate-/l/ 79.4%

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

       4. associate-/l/ 80.2%

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

       5. associate-*r/ 80.2%

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

       6. associate-/l/ 79.6%

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

       7. unpow2 79.6%

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

       8. times-frac 79.7%

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

       9. *-commutative 79.7%

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

       10. times-frac 76.7%

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

       11. unpow2 76.7%

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

       12. unpow2 76.7%

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

       13. times-frac 86.9%

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

       14. unpow2 86.9%

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

   14. Simplified 86.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -230000 \lor \neg \left(t \leq 6.4 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 2: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ 1.0 (+ t_1 1.0))))
        2e+305)
     (/
      (pow
       (* (/ (cbrt (/ 2.0 (tan k))) t) (* (cbrt l) (cbrt (/ l (sin k)))))
       3.0)
      (+ 2.0 t_1))
     (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (cos k) t)) (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 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 2e+305) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * (cbrt(l) * cbrt((l / sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * (cos(k) / t)) / 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 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 2e+305) {
		tmp = Math.pow(((Math.cbrt((2.0 / Math.tan(k))) / t) * (Math.cbrt(l) * Math.cbrt((l / Math.sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * (Math.cos(k) / t)) / 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(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_1 + 1.0)))) <= 2e+305)
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * Float64(cbrt(l) * cbrt(Float64(l / sin(k))))) ^ 3.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / (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[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+305], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1.9999999999999999e305

   1. Initial program 82.7%

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

      1. associate-/r* 82.7%

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

      2. associate-*l* 71.7%

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

      3. sqr-neg 71.7%

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

      4. associate-*l* 82.7%

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

      5. *-commutative 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-/r* 82.7%

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

   3. Simplified 83.2%

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

      1. associate-/l/ 83.2%

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

      2. associate-/r/ 82.7%

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

      3. add-cube-cbrt 82.6%

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

   5. Applied egg-rr 91.0%

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

      1. pow-plus 91.0%

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

      2. metadata-eval 91.0%

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

      3. associate-/r/ 91.2%

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

      4. associate-/r/ 91.0%

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

   7. Simplified 91.0%

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

      1. cbrt-prod 94.7%

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

   9. Applied egg-rr 94.7%

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

   if 1.9999999999999999e305 < (/.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 22.7%

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

      1. associate-/r* 22.7%

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

      2. associate-*l* 22.7%

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

      3. sqr-neg 22.7%

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

      4. associate-*l* 22.7%

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

      5. *-commutative 22.7%

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

      6. sqr-neg 22.7%

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

      7. associate-/r* 22.7%

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

   3. Simplified 22.7%

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

      1. associate-/l/ 22.7%

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

      2. associate-/r/ 22.7%

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

      3. add-cube-cbrt 22.7%

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

   5. Applied egg-rr 39.5%

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

      1. pow-plus 39.5%

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

      2. metadata-eval 39.5%

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

      3. associate-/r/ 39.5%

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

      4. associate-/r/ 39.5%

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

   7. Simplified 39.5%

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

      1. cbrt-prod 51.2%

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

   9. Applied egg-rr 51.2%

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

       1. cbrt-div 54.4%

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

   11. Applied egg-rr 54.4%

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

   12. Taylor expanded in k around inf 64.0%

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

       1. times-frac 63.9%

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

       2. unpow2 63.9%

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

       3. associate-/l/ 64.6%

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

       4. associate-/l/ 64.6%

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

       5. associate-*r/ 64.6%

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

       6. associate-/l/ 63.9%

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

       7. unpow2 63.9%

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

       8. times-frac 64.8%

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

       9. *-commutative 64.8%

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

       10. times-frac 61.0%

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

       11. unpow2 61.0%

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

       12. unpow2 61.0%

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

       13. times-frac 80.9%

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

       14. unpow2 80.9%

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

   14. Simplified 80.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ 1.0 (+ t_1 1.0))))
        5e+298)
     (/
      (pow (* (/ (cbrt (/ 2.0 (tan k))) t) (cbrt (* l (/ l (sin k))))) 3.0)
      (+ 2.0 t_1))
     (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (cos k) t)) (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 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 5e+298) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * cbrt((l * (l / sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * (cos(k) / t)) / 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 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 5e+298) {
		tmp = Math.pow(((Math.cbrt((2.0 / Math.tan(k))) / t) * Math.cbrt((l * (l / Math.sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * (Math.cos(k) / t)) / 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(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_1 + 1.0)))) <= 5e+298)
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * cbrt(Float64(l * Float64(l / sin(k))))) ^ 3.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / (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[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+298], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[Power[N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.0000000000000003e298

   1. Initial program 82.6%

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

      1. associate-/r* 82.6%

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

      2. associate-*l* 72.1%

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

      3. sqr-neg 72.1%

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

      4. associate-*l* 82.6%

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

      5. *-commutative 82.6%

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

      6. sqr-neg 82.6%

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

      7. associate-/r* 82.6%

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

   3. Simplified 83.7%

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

      1. associate-/l/ 83.7%

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

      2. associate-/r/ 82.6%

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

      3. add-cube-cbrt 82.5%

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

   5. Applied egg-rr 91.5%

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

      1. pow-plus 91.6%

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

      2. metadata-eval 91.6%

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

      3. associate-/r/ 91.7%

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

      4. associate-/r/ 91.6%

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

   7. Simplified 91.6%

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

   if 5.0000000000000003e298 < (/.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 23.3%

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

      1. associate-/r* 23.3%

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

      2. associate-*l* 22.6%

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

      3. sqr-neg 22.6%

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

      4. associate-*l* 23.3%

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

      5. *-commutative 23.3%

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

      6. sqr-neg 23.3%

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

      7. associate-/r* 23.3%

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

   3. Simplified 22.6%

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

      1. associate-/l/ 22.6%

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

      2. associate-/r/ 23.3%

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

      3. add-cube-cbrt 23.3%

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

   5. Applied egg-rr 39.2%

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

      1. pow-plus 39.2%

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

      2. metadata-eval 39.2%

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

      3. associate-/r/ 39.2%

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

      4. associate-/r/ 39.2%

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

   7. Simplified 39.2%

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

      1. cbrt-prod 51.6%

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

   9. Applied egg-rr 51.6%

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

       1. cbrt-div 54.8%

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

   11. Applied egg-rr 54.8%

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

   12. Taylor expanded in k around inf 63.5%

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

       1. times-frac 63.5%

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

       2. unpow2 63.5%

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

       3. associate-/l/ 64.1%

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

       4. associate-/l/ 64.1%

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

       5. associate-*r/ 64.1%

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

       6. associate-/l/ 63.5%

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

       7. unpow2 63.5%

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

       8. times-frac 64.3%

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

       9. *-commutative 64.3%

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

       10. times-frac 60.6%

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

       11. unpow2 60.6%

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

       12. unpow2 60.6%

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

       13. times-frac 80.3%

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

       14. unpow2 80.3%

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

   14. Simplified 80.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 4: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ 1.0 (+ t_1 1.0))))
        5e+298)
     (/
      (pow (* (/ (cbrt (/ 2.0 (tan k))) t) (cbrt (/ l (/ (sin k) l)))) 3.0)
      (+ 2.0 t_1))
     (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (cos k) t)) (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 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 5e+298) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * cbrt((l / (sin(k) / l)))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * (cos(k) / t)) / 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 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 5e+298) {
		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 * ((Math.pow((l / k), 2.0) * (Math.cos(k) / t)) / 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(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_1 + 1.0)))) <= 5e+298)
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * cbrt(Float64(l / Float64(sin(k) / l)))) ^ 3.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / (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[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+298], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[Power[N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.0000000000000003e298

   1. Initial program 82.6%

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

      1. associate-/r* 82.6%

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

      2. associate-*l* 72.1%

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

      3. sqr-neg 72.1%

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

      4. associate-*l* 82.6%

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

      5. *-commutative 82.6%

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

      6. sqr-neg 82.6%

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

      7. associate-/r* 82.6%

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

   3. Simplified 83.7%

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

      1. associate-/l/ 83.7%

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

      2. associate-/r/ 82.6%

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

      3. add-cube-cbrt 82.5%

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

   5. Applied egg-rr 91.5%

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

      1. pow-plus 91.6%

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

      2. metadata-eval 91.6%

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

      3. associate-/r/ 91.7%

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

      4. associate-/r/ 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in k around inf 57.6%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{{\left(\frac{{\ell}^{2}}{\sin k}\right)}^{0.3333333333333333}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow1/3 86.9%

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

      2. unpow2 86.9%

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

      3. associate-/l* 91.7%

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

   10. Simplified 91.7%

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

   if 5.0000000000000003e298 < (/.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 23.3%

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

      1. associate-/r* 23.3%

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

      2. associate-*l* 22.6%

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

      3. sqr-neg 22.6%

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

      4. associate-*l* 23.3%

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

      5. *-commutative 23.3%

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

      6. sqr-neg 23.3%

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

      7. associate-/r* 23.3%

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

   3. Simplified 22.6%

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

      1. associate-/l/ 22.6%

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

      2. associate-/r/ 23.3%

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

      3. add-cube-cbrt 23.3%

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

   5. Applied egg-rr 39.2%

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

      1. pow-plus 39.2%

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

      2. metadata-eval 39.2%

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

      3. associate-/r/ 39.2%

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

      4. associate-/r/ 39.2%

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

   7. Simplified 39.2%

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

      1. cbrt-prod 51.6%

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

   9. Applied egg-rr 51.6%

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

       1. cbrt-div 54.8%

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

   11. Applied egg-rr 54.8%

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

   12. Taylor expanded in k around inf 63.5%

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

       1. times-frac 63.5%

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

       2. unpow2 63.5%

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

       3. associate-/l/ 64.1%

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

       4. associate-/l/ 64.1%

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

       5. associate-*r/ 64.1%

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

       6. associate-/l/ 63.5%

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

       7. unpow2 63.5%

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

       8. times-frac 64.3%

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

       9. *-commutative 64.3%

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

       10. times-frac 60.6%

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

       11. unpow2 60.6%

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

       12. unpow2 60.6%

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

       13. times-frac 80.3%

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

       14. unpow2 80.3%

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

   14. Simplified 80.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 5: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)} \cdot \frac{1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ 1.0 (+ t_1 1.0))))
        2e+305)
     (*
      (/ 2.0 (* (tan k) (* (/ (sin k) l) (/ (pow t 3.0) l))))
      (/ 1.0 (+ 2.0 t_1)))
     (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (cos k) t)) (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 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 2e+305) {
		tmp = (2.0 / (tan(k) * ((sin(k) / l) * (pow(t, 3.0) / l)))) * (1.0 / (2.0 + t_1));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))) * (1.0d0 + (t_1 + 1.0d0)))) <= 2d+305) then
        tmp = (2.0d0 / (tan(k) * ((sin(k) / l) * ((t ** 3.0d0) / l)))) * (1.0d0 / (2.0d0 + t_1))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) * (cos(k) / t)) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 2e+305) {
		tmp = (2.0 / (Math.tan(k) * ((Math.sin(k) / l) * (Math.pow(t, 3.0) / l)))) * (1.0 / (2.0 + t_1));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * (Math.cos(k) / t)) / 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 / ((math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 2e+305:
		tmp = (2.0 / (math.tan(k) * ((math.sin(k) / l) * (math.pow(t, 3.0) / l)))) * (1.0 / (2.0 + t_1))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) * (math.cos(k) / t)) / 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(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_1 + 1.0)))) <= 2e+305)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64(sin(k) / l) * Float64((t ^ 3.0) / l)))) * Float64(1.0 / Float64(2.0 + t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / (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 / ((tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))) * (1.0 + (t_1 + 1.0)))) <= 2e+305)
		tmp = (2.0 / (tan(k) * ((sin(k) / l) * ((t ^ 3.0) / l)))) * (1.0 / (2.0 + t_1));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) * (cos(k) / t)) / (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[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+305], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1.9999999999999999e305

   1. Initial program 82.7%

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

      1. associate-/r* 82.7%

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

      2. +-commutative 82.7%

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

   3. Simplified 82.7%

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

      1. add-sqr-sqrt 41.5%

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

      2. pow2 41.5%

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

      3. sqrt-div 41.5%

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

      4. sqrt-pow1 42.2%

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

      5. metadata-eval 42.2%

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

      6. sqrt-prod 18.4%

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

      7. add-sqr-sqrt 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. div-inv 43.5%

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

      2. *-commutative 43.5%

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

      3. *-commutative 43.5%

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

      4. associate-+r+ 43.5%

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

      5. metadata-eval 43.5%

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

   7. Applied egg-rr 43.5%

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

   8. Taylor expanded in k around inf 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 89.3%

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

   10. Simplified 89.3%

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

   if 1.9999999999999999e305 < (/.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 22.7%

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

      1. associate-/r* 22.7%

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

      2. associate-*l* 22.7%

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

      3. sqr-neg 22.7%

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

      4. associate-*l* 22.7%

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

      5. *-commutative 22.7%

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

      6. sqr-neg 22.7%

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

      7. associate-/r* 22.7%

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

   3. Simplified 22.7%

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

      1. associate-/l/ 22.7%

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

      2. associate-/r/ 22.7%

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

      3. add-cube-cbrt 22.7%

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

   5. Applied egg-rr 39.5%

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

      1. pow-plus 39.5%

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

      2. metadata-eval 39.5%

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

      3. associate-/r/ 39.5%

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

      4. associate-/r/ 39.5%

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

   7. Simplified 39.5%

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

      1. cbrt-prod 51.2%

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

   9. Applied egg-rr 51.2%

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

       1. cbrt-div 54.4%

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

   11. Applied egg-rr 54.4%

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

   12. Taylor expanded in k around inf 64.0%

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

       1. times-frac 63.9%

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

       2. unpow2 63.9%

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

       3. associate-/l/ 64.6%

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

       4. associate-/l/ 64.6%

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

       5. associate-*r/ 64.6%

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

       6. associate-/l/ 63.9%

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

       7. unpow2 63.9%

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

       8. times-frac 64.8%

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

       9. *-commutative 64.8%

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

       10. times-frac 61.0%

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

       11. unpow2 61.0%

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

       12. unpow2 61.0%

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

       13. times-frac 80.9%

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

       14. unpow2 80.9%

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

   14. Simplified 80.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 6: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 2.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t_1} \cdot \frac{\ell}{t}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 2.3e-15)
     (* (/ l k) (/ l (* k (pow t 3.0))))
     (if (<= k 7.2e+62)
       (* 2.0 (/ (* (cos k) (* (/ l t_1) (/ l t))) (* k k)))
       (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (cos k) t)) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 2.3e-15) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else if (k <= 7.2e+62) {
		tmp = 2.0 * ((cos(k) * ((l / t_1) * (l / t))) / (k * k));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * (cos(k) / t)) / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 2.3d-15) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else if (k <= 7.2d+62) then
        tmp = 2.0d0 * ((cos(k) * ((l / t_1) * (l / t))) / (k * k))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) * (cos(k) / t)) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 2.3e-15) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else if (k <= 7.2e+62) {
		tmp = 2.0 * ((Math.cos(k) * ((l / t_1) * (l / t))) / (k * k));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * (Math.cos(k) / t)) / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 2.3e-15:
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	elif k <= 7.2e+62:
		tmp = 2.0 * ((math.cos(k) * ((l / t_1) * (l / t))) / (k * k))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) * (math.cos(k) / t)) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 2.3e-15)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	elseif (k <= 7.2e+62)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / t_1) * Float64(l / t))) / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / t)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 2.3e-15)
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	elseif (k <= 7.2e+62)
		tmp = 2.0 * ((cos(k) * ((l / t_1) * (l / t))) / (k * k));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) * (cos(k) / t)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 2.3e-15], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.2e+62], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / t$95$1), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.2999999999999999e-15

   1. Initial program 61.7%

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

      1. associate-/r* 61.7%

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

      2. associate-*l* 53.1%

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

      3. sqr-neg 53.1%

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

      4. associate-*l* 61.7%

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

      5. *-commutative 61.7%

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

      6. sqr-neg 61.7%

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

      7. associate-/r* 61.7%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 62.1%

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

      2. associate-/r/ 61.7%

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

      3. add-cube-cbrt 61.7%

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

   5. Applied egg-rr 74.5%

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

      1. pow-plus 74.5%

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

      2. metadata-eval 74.5%

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

      3. associate-/r/ 74.6%

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

      4. associate-/r/ 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in k around 0 53.9%

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

      1. unpow2 53.9%

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

      2. associate-*r* 62.0%

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

      3. unpow2 62.0%

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

      4. times-frac 68.3%

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

   10. Simplified 68.3%

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

   if 2.2999999999999999e-15 < k < 7.2e62

   1. Initial program 57.6%

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

      1. associate-/r* 57.8%

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

      2. associate-*l* 57.8%

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

      3. sqr-neg 57.8%

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

      4. associate-*l* 57.8%

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

      5. *-commutative 57.8%

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

      6. sqr-neg 57.8%

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

      7. associate-*l/ 57.7%

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

      8. associate-*r/ 57.8%

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

      9. associate-/r/ 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in k around inf 74.6%

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

      1. times-frac 74.6%

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

      2. unpow2 74.6%

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

      3. unpow2 74.6%

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

      4. times-frac 84.5%

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

   6. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

   8. Applied egg-rr 84.5%

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

   if 7.2e62 < k

   1. Initial program 37.1%

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

      1. associate-/r* 37.1%

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

      2. associate-*l* 37.1%

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

      3. sqr-neg 37.1%

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

      4. associate-*l* 37.1%

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

      5. *-commutative 37.1%

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

      6. sqr-neg 37.1%

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

      7. associate-/r* 37.1%

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

   3. Simplified 37.1%

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

      1. associate-/l/ 37.1%

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

      2. associate-/r/ 37.1%

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

      3. add-cube-cbrt 37.1%

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

   5. Applied egg-rr 48.8%

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

      1. pow-plus 48.8%

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

      2. metadata-eval 48.8%

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

      3. associate-/r/ 48.9%

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

      4. associate-/r/ 48.9%

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

   7. Simplified 48.9%

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

      1. cbrt-prod 61.7%

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

   9. Applied egg-rr 61.7%

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

       1. cbrt-div 61.6%

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

   11. Applied egg-rr 61.6%

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

   12. Taylor expanded in k around inf 60.3%

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

       1. times-frac 54.8%

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

       2. unpow2 54.8%

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

       3. associate-/l/ 56.2%

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

       4. associate-/l/ 56.2%

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

       5. associate-*r/ 56.2%

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

       6. associate-/l/ 54.8%

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

       7. unpow2 54.8%

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

       8. times-frac 60.2%

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

       9. *-commutative 60.2%

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

       10. times-frac 53.4%

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

       11. unpow2 53.4%

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

       12. unpow2 53.4%

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

       13. times-frac 88.1%

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

       14. unpow2 88.1%

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

   14. Simplified 88.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 7: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -120000 \lor \neg \left(t \leq 5.5 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -120000.0) (not (<= t 5.5e-86)))
   (/
    (/ 2.0 (* (tan k) (* (/ (pow t 3.0) l) (/ k l))))
    (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
   (* 2.0 (/ (* (cos k) (* (/ l (pow (sin k) 2.0)) (/ l t))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -120000.0) || !(t <= 5.5e-86)) {
		tmp = (2.0 / (tan(k) * ((pow(t, 3.0) / l) * (k / l)))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / pow(sin(k), 2.0)) * (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 <= (-120000.0d0)) .or. (.not. (t <= 5.5d-86))) then
        tmp = (2.0d0 / (tan(k) * (((t ** 3.0d0) / l) * (k / l)))) / (1.0d0 + (((k / t) ** 2.0d0) + 1.0d0))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / (sin(k) ** 2.0d0)) * (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 <= -120000.0) || !(t <= 5.5e-86)) {
		tmp = (2.0 / (Math.tan(k) * ((Math.pow(t, 3.0) / l) * (k / l)))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / Math.pow(Math.sin(k), 2.0)) * (l / t))) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -120000.0) or not (t <= 5.5e-86):
		tmp = (2.0 / (math.tan(k) * ((math.pow(t, 3.0) / l) * (k / l)))) / (1.0 + (math.pow((k / t), 2.0) + 1.0))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / math.pow(math.sin(k), 2.0)) * (l / t))) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -120000.0) || !(t <= 5.5e-86))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64((t ^ 3.0) / l) * Float64(k / l)))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(l / t))) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -120000.0) || ~((t <= 5.5e-86)))
		tmp = (2.0 / (tan(k) * (((t ^ 3.0) / l) * (k / l)))) / (1.0 + (((k / t) ^ 2.0) + 1.0));
	else
		tmp = 2.0 * ((cos(k) * ((l / (sin(k) ^ 2.0)) * (l / t))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -120000.0], N[Not[LessEqual[t, 5.5e-86]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2e5 or 5.5e-86 < t

   1. Initial program 69.6%

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

      1. associate-/r* 69.6%

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

      2. +-commutative 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in k around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. unpow2 68.0%

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

      3. times-frac 77.2%

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

   6. Simplified 77.2%

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

   if -1.2e5 < t < 5.5e-86

   1. Initial program 40.3%

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

      1. associate-/r* 40.3%

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

      2. associate-*l* 40.3%

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

      3. sqr-neg 40.3%

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

      4. associate-*l* 40.3%

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

      5. *-commutative 40.3%

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

      6. sqr-neg 40.3%

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

      7. associate-*l/ 40.3%

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

      8. associate-*r/ 40.3%

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

      9. associate-/r/ 40.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in k around inf 79.4%

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

      1. times-frac 79.3%

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

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

      4. times-frac 85.3%

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

   6. Simplified 85.3%

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

      1. associate-*l/ 85.3%

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

   8. Applied egg-rr 85.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -120000 \lor \neg \left(t \leq 5.5 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}{k \cdot k}\\ \end{array} \]

Alternative 8: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -165000 \lor \neg \left(t \leq 1.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -165000.0) (not (<= t 1.5e-23)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* 2.0 (* (* (/ l (pow (sin k) 2.0)) (/ l t)) (/ (cos k) (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -165000.0) || !(t <= 1.5e-23)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((l / pow(sin(k), 2.0)) * (l / t)) * (cos(k) / (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 <= (-165000.0d0)) .or. (.not. (t <= 1.5d-23))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * (((l / (sin(k) ** 2.0d0)) * (l / t)) * (cos(k) / (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -165000.0) || !(t <= 1.5e-23)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (((l / Math.pow(Math.sin(k), 2.0)) * (l / t)) * (Math.cos(k) / (k * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -165000.0) or not (t <= 1.5e-23):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (((l / math.pow(math.sin(k), 2.0)) * (l / t)) * (math.cos(k) / (k * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -165000.0) || !(t <= 1.5e-23))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(l / t)) * Float64(cos(k) / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -165000.0) || ~((t <= 1.5e-23)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * (((l / (sin(k) ^ 2.0)) * (l / t)) * (cos(k) / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -165000.0], N[Not[LessEqual[t, 1.5e-23]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -165000 or 1.50000000000000001e-23 < t

   1. Initial program 68.4%

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

      1. associate-/r* 68.4%

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

      2. associate-*l* 55.6%

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

      3. sqr-neg 55.6%

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

      4. associate-*l* 68.4%

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

      5. *-commutative 68.4%

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

      6. sqr-neg 68.4%

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

      7. associate-/r* 68.4%

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

   3. Simplified 69.0%

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

      1. associate-/l/ 69.0%

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

      2. associate-/r/ 68.4%

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

      3. add-cube-cbrt 68.3%

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

   5. Applied egg-rr 77.1%

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

      1. pow-plus 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/r/ 77.2%

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

      4. associate-/r/ 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in k around 0 54.0%

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

      1. unpow2 54.0%

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

      2. associate-*r* 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 77.0%

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

   10. Simplified 77.0%

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

   if -165000 < t < 1.50000000000000001e-23

   1. Initial program 45.2%

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

      1. associate-/r* 45.2%

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

      2. associate-*l* 45.2%

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

      3. sqr-neg 45.2%

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

      4. associate-*l* 45.2%

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

      5. *-commutative 45.2%

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

      6. sqr-neg 45.2%

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

      7. associate-*l/ 44.5%

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

      8. associate-*r/ 44.4%

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

      9. associate-/r/ 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in k around inf 79.2%

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

      1. times-frac 79.1%

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

      2. unpow2 79.1%

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

      3. unpow2 79.1%

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

      4. times-frac 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -165000 \lor \neg \left(t \leq 1.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \]

Alternative 9: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -100000 \lor \neg \left(t \leq 1.72 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -100000.0) (not (<= t 1.72e-23)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* 2.0 (/ (* (cos k) (* (/ l (pow (sin k) 2.0)) (/ l t))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -100000.0) || !(t <= 1.72e-23)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / pow(sin(k), 2.0)) * (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 <= (-100000.0d0)) .or. (.not. (t <= 1.72d-23))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / (sin(k) ** 2.0d0)) * (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 <= -100000.0) || !(t <= 1.72e-23)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / Math.pow(Math.sin(k), 2.0)) * (l / t))) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -100000.0) or not (t <= 1.72e-23):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / math.pow(math.sin(k), 2.0)) * (l / t))) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -100000.0) || !(t <= 1.72e-23))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(l / t))) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -100000.0) || ~((t <= 1.72e-23)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((cos(k) * ((l / (sin(k) ^ 2.0)) * (l / t))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -100000.0], N[Not[LessEqual[t, 1.72e-23]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1e5 or 1.7200000000000001e-23 < t

   1. Initial program 68.4%

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

      1. associate-/r* 68.4%

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

      2. associate-*l* 55.6%

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

      3. sqr-neg 55.6%

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

      4. associate-*l* 68.4%

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

      5. *-commutative 68.4%

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

      6. sqr-neg 68.4%

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

      7. associate-/r* 68.4%

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

   3. Simplified 69.0%

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

      1. associate-/l/ 69.0%

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

      2. associate-/r/ 68.4%

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

      3. add-cube-cbrt 68.3%

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

   5. Applied egg-rr 77.1%

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

      1. pow-plus 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/r/ 77.2%

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

      4. associate-/r/ 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in k around 0 54.0%

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

      1. unpow2 54.0%

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

      2. associate-*r* 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 77.0%

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

   10. Simplified 77.0%

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

   if -1e5 < t < 1.7200000000000001e-23

   1. Initial program 45.2%

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

      1. associate-/r* 45.2%

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

      2. associate-*l* 45.2%

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

      3. sqr-neg 45.2%

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

      4. associate-*l* 45.2%

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

      5. *-commutative 45.2%

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

      6. sqr-neg 45.2%

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

      7. associate-*l/ 44.5%

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

      8. associate-*r/ 44.4%

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

      9. associate-/r/ 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in k around inf 79.2%

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

      1. times-frac 79.1%

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

      2. unpow2 79.1%

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

      3. unpow2 79.1%

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

      4. times-frac 84.4%

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

   6. Simplified 84.4%

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

      1. associate-*l/ 84.4%

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

   8. Applied egg-rr 84.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -100000 \lor \neg \left(t \leq 1.72 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}{k \cdot k}\\ \end{array} \]

Alternative 10: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -100000 \lor \neg \left(t \leq 1.65 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -100000.0) (not (<= t 1.65e-23)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (*
    2.0
    (*
     (pow (/ l k) 2.0)
     (/ (* (cos k) (+ 0.3333333333333333 (/ 1.0 (* k k)))) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -100000.0) || !(t <= 1.65e-23)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * ((cos(k) * (0.3333333333333333 + (1.0 / (k * k)))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-100000.0d0)) .or. (.not. (t <= 1.65d-23))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * ((cos(k) * (0.3333333333333333d0 + (1.0d0 / (k * k)))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -100000.0) || !(t <= 1.65e-23)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * ((Math.cos(k) * (0.3333333333333333 + (1.0 / (k * k)))) / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -100000.0) or not (t <= 1.65e-23):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * ((math.cos(k) * (0.3333333333333333 + (1.0 / (k * k)))) / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -100000.0) || !(t <= 1.65e-23))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(Float64(cos(k) * Float64(0.3333333333333333 + Float64(1.0 / Float64(k * k)))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -100000.0) || ~((t <= 1.65e-23)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * ((cos(k) * (0.3333333333333333 + (1.0 / (k * k)))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -100000.0], N[Not[LessEqual[t, 1.65e-23]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(0.3333333333333333 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1e5 or 1.6500000000000001e-23 < t

   1. Initial program 68.4%

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

      1. associate-/r* 68.4%

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

      2. associate-*l* 55.6%

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

      3. sqr-neg 55.6%

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

      4. associate-*l* 68.4%

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

      5. *-commutative 68.4%

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

      6. sqr-neg 68.4%

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

      7. associate-/r* 68.4%

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

   3. Simplified 69.0%

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

      1. associate-/l/ 69.0%

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

      2. associate-/r/ 68.4%

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

      3. add-cube-cbrt 68.3%

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

   5. Applied egg-rr 77.1%

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

      1. pow-plus 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/r/ 77.2%

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

      4. associate-/r/ 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in k around 0 54.0%

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

      1. unpow2 54.0%

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

      2. associate-*r* 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 77.0%

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

   10. Simplified 77.0%

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

   if -1e5 < t < 1.6500000000000001e-23

   1. Initial program 45.2%

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

      1. associate-/r* 45.2%

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

      2. associate-*l* 45.2%

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

      3. sqr-neg 45.2%

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

      4. associate-*l* 45.2%

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

      5. *-commutative 45.2%

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

      6. sqr-neg 45.2%

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

      7. associate-*l/ 44.5%

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

      8. associate-*r/ 44.4%

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

      9. associate-/r/ 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in k around inf 79.2%

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

      1. times-frac 79.1%

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

      2. unpow2 79.1%

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

      3. unpow2 79.1%

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

      4. times-frac 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in k around 0 69.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 69.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      2. fma-def 69.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      3. unpow2 69.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      4. associate-*r/ 69.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      5. unpow2 69.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right)\right) \]

      6. associate-/l* 72.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot t}{\ell}}}\right)\right) \]

      7. unpow2 72.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell}}\right)\right) \]

   9. Simplified 72.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \frac{\ell}{\frac{\left(k \cdot k\right) \cdot t}{\ell}}\right)}\right) \]

   10. Taylor expanded in t around -inf 69.9%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 69.9%

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

       2. times-frac 70.2%

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

       3. distribute-rgt-neg-in 70.2%

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

       4. mul-1-neg 70.2%

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

       5. unsub-neg 70.2%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{-0.3333333333333333 \cdot {\ell}^{2} - \frac{{\ell}^{2}}{{k}^{2}}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       6. *-commutative 70.2%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -0.3333333333333333} - \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       7. unpow2 70.2%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.3333333333333333 - \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       8. associate-*l* 70.2%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)} - \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       9. unpow2 70.2%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       10. unpow2 70.2%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       11. times-frac 72.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       12. unpow2 72.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{k \cdot k}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

   12. Simplified 72.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \cdot \left(-\frac{\cos k}{t}\right)\right)} \]

   13. Taylor expanded in l around 0 69.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(0.3333333333333333 + \frac{1}{{k}^{2}}\right) \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot t}} \]
   14. Step-by-step derivation

       1. associate-*r* 69.9%

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

       2. *-commutative 69.9%

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

       3. times-frac 69.5%

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

       4. unpow2 69.5%

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

       5. unpow2 69.5%

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

       6. unpow2 69.5%

          \[\leadsto 2 \cdot \left(\frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \cos k}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

       7. times-frac 73.6%

          \[\leadsto 2 \cdot \left(\frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \cos k}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

       8. unpow2 73.6%

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

   15. Simplified 73.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \cos k}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -100000 \lor \neg \left(t \leq 1.65 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k \cdot \left(0.3333333333333333 + \frac{1}{k \cdot k}\right)}{t}\right)\\ \end{array} \]

Alternative 11: 71.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -100000 \lor \neg \left(t \leq 1.65 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -100000.0) (not (<= t 1.65e-23)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (*
    2.0
    (*
     (/ (cos k) (* k k))
     (* (/ l t) (+ (/ l (* k k)) (* l 0.3333333333333333)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -100000.0) || !(t <= 1.65e-23)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	}
	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 <= (-100000.0d0)) .or. (.not. (t <= 1.65d-23))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -100000.0) || !(t <= 1.65e-23)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -100000.0) or not (t <= 1.65e-23):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -100000.0) || !(t <= 1.65e-23))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(Float64(l / Float64(k * k)) + Float64(l * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -100000.0) || ~((t <= 1.65e-23)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -100000.0], N[Not[LessEqual[t, 1.65e-23]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1e5 or 1.6500000000000001e-23 < t

   1. Initial program 68.4%

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

      1. associate-/r* 68.4%

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

      2. associate-*l* 55.6%

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

      3. sqr-neg 55.6%

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

      4. associate-*l* 68.4%

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

      5. *-commutative 68.4%

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

      6. sqr-neg 68.4%

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

      7. associate-/r* 68.4%

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

   3. Simplified 69.0%

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

      1. associate-/l/ 69.0%

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

      2. associate-/r/ 68.4%

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

      3. add-cube-cbrt 68.3%

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

   5. Applied egg-rr 77.1%

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

      1. pow-plus 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/r/ 77.2%

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

      4. associate-/r/ 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in k around 0 54.0%

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

      1. unpow2 54.0%

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

      2. associate-*r* 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 77.0%

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

   10. Simplified 77.0%

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

   if -1e5 < t < 1.6500000000000001e-23

   1. Initial program 45.2%

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

      1. associate-/r* 45.2%

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

      2. associate-*l* 45.2%

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

      3. sqr-neg 45.2%

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

      4. associate-*l* 45.2%

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

      5. *-commutative 45.2%

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

      6. sqr-neg 45.2%

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

      7. associate-*l/ 44.5%

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

      8. associate-*r/ 44.4%

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

      9. associate-/r/ 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in k around inf 79.2%

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

      1. times-frac 79.1%

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

      2. unpow2 79.1%

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

      3. unpow2 79.1%

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

      4. times-frac 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in k around 0 73.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)} \cdot \frac{\ell}{t}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 73.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\frac{\ell}{\color{blue}{k \cdot k}} + 0.3333333333333333 \cdot \ell\right) \cdot \frac{\ell}{t}\right)\right) \]

      2. *-commutative 73.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\frac{\ell}{k \cdot k} + \color{blue}{\ell \cdot 0.3333333333333333}\right) \cdot \frac{\ell}{t}\right)\right) \]

   9. Simplified 73.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\color{blue}{\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)} \cdot \frac{\ell}{t}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -100000 \lor \neg \left(t \leq 1.65 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]

Alternative 12: 71.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{2 \cdot t_1}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l (* k (pow t 3.0))))))
   (if (<= t -1.2e-72)
     (/ (* 2.0 t_1) (+ 2.0 (* (/ k t) (/ k t))))
     (if (<= t 1.5e-23)
       (*
        2.0
        (*
         (/ (cos k) (* k k))
         (* (/ l t) (+ (/ l (* k k)) (* l 0.3333333333333333)))))
       t_1))))

C

double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / (k * pow(t, 3.0)));
	double tmp;
	if (t <= -1.2e-72) {
		tmp = (2.0 * t_1) / (2.0 + ((k / t) * (k / t)));
	} else if (t <= 1.5e-23) {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) * (l / (k * (t ** 3.0d0)))
    if (t <= (-1.2d-72)) then
        tmp = (2.0d0 * t_1) / (2.0d0 + ((k / t) * (k / t)))
    else if (t <= 1.5d-23) then
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / (k * Math.pow(t, 3.0)));
	double tmp;
	if (t <= -1.2e-72) {
		tmp = (2.0 * t_1) / (2.0 + ((k / t) * (k / t)));
	} else if (t <= 1.5e-23) {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / k) * (l / (k * math.pow(t, 3.0)))
	tmp = 0
	if t <= -1.2e-72:
		tmp = (2.0 * t_1) / (2.0 + ((k / t) * (k / t)))
	elif t <= 1.5e-23:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))))
	tmp = 0.0
	if (t <= -1.2e-72)
		tmp = Float64(Float64(2.0 * t_1) / Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))));
	elseif (t <= 1.5e-23)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(Float64(l / Float64(k * k)) + Float64(l * 0.3333333333333333)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) * (l / (k * (t ^ 3.0)));
	tmp = 0.0;
	if (t <= -1.2e-72)
		tmp = (2.0 * t_1) / (2.0 + ((k / t) * (k / t)));
	elseif (t <= 1.5e-23)
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2e-72], N[(N[(2.0 * t$95$1), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e-23], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2e-72

   1. Initial program 68.3%

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

      1. associate-/r* 68.3%

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

      2. associate-*l* 61.8%

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

      3. sqr-neg 61.8%

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

      4. associate-*l* 68.3%

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

      5. *-commutative 68.3%

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

      6. sqr-neg 68.3%

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

      7. associate-/r* 68.3%

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

   3. Simplified 68.6%

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

      1. expm1-log1p-u 52.1%

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

      2. expm1-udef 48.8%

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

      3. associate-/l* 48.8%

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

   5. Applied egg-rr 48.8%

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

      1. expm1-def 58.0%

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

      2. expm1-log1p 74.5%

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

      3. associate-/r/ 74.5%

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

   7. Simplified 74.5%

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

      1. unpow2 74.5%

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

   9. Applied egg-rr 74.5%

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

   10. Taylor expanded in k around 0 57.2%

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

       1. unpow2 57.2%

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

       2. associate-*r* 63.5%

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

       3. unpow2 63.5%

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

       4. times-frac 72.0%

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

   12. Simplified 72.0%

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

   if -1.2e-72 < t < 1.50000000000000001e-23

   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* 40.6%

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

      2. associate-*l* 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* 40.6%

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

      5. *-commutative 40.6%

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

      6. sqr-neg 40.6%

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

      7. associate-*l/ 39.7%

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

      8. associate-*r/ 39.7%

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

      9. associate-/r/ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in k around inf 79.6%

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

      1. times-frac 79.4%

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

      2. unpow2 79.4%

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

      3. unpow2 79.4%

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

      4. times-frac 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in k around 0 75.1%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)} \cdot \frac{\ell}{t}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 75.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\frac{\ell}{\color{blue}{k \cdot k}} + 0.3333333333333333 \cdot \ell\right) \cdot \frac{\ell}{t}\right)\right) \]

      2. *-commutative 75.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\left(\frac{\ell}{k \cdot k} + \color{blue}{\ell \cdot 0.3333333333333333}\right) \cdot \frac{\ell}{t}\right)\right) \]

   9. Simplified 75.1%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\color{blue}{\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)} \cdot \frac{\ell}{t}\right)\right) \]

   if 1.50000000000000001e-23 < 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* 70.3%

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

      2. associate-*l* 55.2%

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

      3. sqr-neg 55.2%

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

      4. associate-*l* 70.3%

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

      5. *-commutative 70.3%

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

      6. sqr-neg 70.3%

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

      7. associate-/r* 70.3%

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

   3. Simplified 70.9%

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

      1. associate-/l/ 70.9%

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

      2. associate-/r/ 70.3%

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

      3. add-cube-cbrt 70.2%

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

   5. Applied egg-rr 76.5%

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

      1. pow-plus 76.6%

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

      2. metadata-eval 76.6%

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

      3. associate-/r/ 76.8%

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

      4. associate-/r/ 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in k around 0 52.7%

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

      1. unpow2 52.7%

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

      2. associate-*r* 66.7%

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

      3. unpow2 66.7%

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

      4. times-frac 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 13: 71.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -720 \lor \neg \left(t \leq 1.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -720.0) (not (<= t 1.5e-23)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l t) (/ l (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -720.0) || !(t <= 1.5e-23)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (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 <= (-720.0d0)) .or. (.not. (t <= 1.5d-23))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -720.0) || !(t <= 1.5e-23)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -720.0) or not (t <= 1.5e-23):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -720.0) || !(t <= 1.5e-23))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -720.0) || ~((t <= 1.5e-23)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -720.0], N[Not[LessEqual[t, 1.5e-23]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -720 or 1.50000000000000001e-23 < t

   1. Initial program 68.9%

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

      1. associate-/r* 68.9%

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

      2. associate-*l* 56.3%

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

      3. sqr-neg 56.3%

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

      4. associate-*l* 68.9%

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

      5. *-commutative 68.9%

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

      6. sqr-neg 68.9%

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

      7. associate-/r* 68.9%

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

   3. Simplified 69.4%

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

      1. associate-/l/ 69.4%

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

      2. associate-/r/ 68.9%

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

      3. add-cube-cbrt 68.8%

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

   5. Applied egg-rr 77.4%

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

      1. pow-plus 77.4%

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

      2. metadata-eval 77.4%

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

      3. associate-/r/ 77.5%

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

      4. associate-/r/ 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in k around 0 53.2%

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

      1. unpow2 53.2%

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

      2. associate-*r* 65.1%

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

      3. unpow2 65.1%

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

      4. times-frac 75.9%

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

   10. Simplified 75.9%

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

   if -720 < t < 1.50000000000000001e-23

   1. Initial program 44.4%

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

      1. associate-/r* 44.4%

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

      2. associate-*l* 44.4%

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

      3. sqr-neg 44.4%

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

      4. associate-*l* 44.4%

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

      5. *-commutative 44.4%

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

      6. sqr-neg 44.4%

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

      7. associate-*l/ 43.6%

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

      8. associate-*r/ 43.6%

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

      9. associate-/r/ 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in k around inf 78.9%

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

      1. times-frac 78.8%

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

      2. unpow2 78.8%

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

      3. unpow2 78.8%

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

      4. times-frac 84.2%

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

   6. Simplified 84.2%

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

   7. Taylor expanded in k around 0 72.8%

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

      1. unpow2 72.8%

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

   9. Simplified 72.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -720 \lor \neg \left(t \leq 1.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 14: 65.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.3e-14)
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* 2.0 (* (/ (cos k) (* k k)) (/ l (/ (* t (* k k)) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-14) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (l / ((t * (k * k)) / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.3d-14) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (l / ((t * (k * k)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-14) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (l / ((t * (k * k)) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.3e-14:
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (l / ((t * (k * k)) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.3e-14)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(l / Float64(Float64(t * Float64(k * k)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.3e-14)
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * (l / ((t * (k * k)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.3e-14], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.29999999999999998e-14

   1. Initial program 61.7%

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

      1. associate-/r* 61.7%

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

      2. associate-*l* 53.1%

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

      3. sqr-neg 53.1%

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

      4. associate-*l* 61.7%

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

      5. *-commutative 61.7%

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

      6. sqr-neg 61.7%

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

      7. associate-/r* 61.7%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 62.1%

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

      2. associate-/r/ 61.7%

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

      3. add-cube-cbrt 61.7%

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

   5. Applied egg-rr 74.5%

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

      1. pow-plus 74.5%

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

      2. metadata-eval 74.5%

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

      3. associate-/r/ 74.6%

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

      4. associate-/r/ 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in k around 0 53.9%

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

      1. unpow2 53.9%

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

      2. associate-*r* 62.0%

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

      3. unpow2 62.0%

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

      4. times-frac 68.3%

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

   10. Simplified 68.3%

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

   if 2.29999999999999998e-14 < k

   1. Initial program 42.6%

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

      1. associate-/r* 42.6%

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

      2. associate-*l* 42.6%

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

      3. sqr-neg 42.6%

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

      4. associate-*l* 42.6%

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

      5. *-commutative 42.6%

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

      6. sqr-neg 42.6%

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

      7. associate-*l/ 42.6%

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

      8. associate-*r/ 42.6%

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

      9. associate-/r/ 42.6%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in k around inf 64.1%

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

      1. times-frac 60.1%

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

      2. unpow2 60.1%

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

      3. unpow2 60.1%

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

      4. times-frac 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in k around 0 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/l* 61.1%

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

      3. unpow2 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)\\ \end{array} \]

Alternative 15: 64.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-11)
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* 2.0 (* 0.3333333333333333 (* (/ (cos k) (* k k)) (* l (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-11) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) / (k * k)) * (l * (l / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-11) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * ((cos(k) / (k * k)) * (l * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-11) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((Math.cos(k) / (k * k)) * (l * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5e-11:
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (0.3333333333333333 * ((math.cos(k) / (k * k)) * (l * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-11)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(l * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-11)
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) / (k * k)) * (l * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5e-11], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000018e-11

   1. Initial program 61.7%

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

      1. associate-/r* 61.7%

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

      2. associate-*l* 53.1%

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

      3. sqr-neg 53.1%

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

      4. associate-*l* 61.7%

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

      5. *-commutative 61.7%

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

      6. sqr-neg 61.7%

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

      7. associate-/r* 61.7%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 62.1%

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

      2. associate-/r/ 61.7%

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

      3. add-cube-cbrt 61.7%

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

   5. Applied egg-rr 74.5%

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

      1. pow-plus 74.5%

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

      2. metadata-eval 74.5%

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

      3. associate-/r/ 74.6%

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

      4. associate-/r/ 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in k around 0 53.9%

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

      1. unpow2 53.9%

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

      2. associate-*r* 62.0%

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

      3. unpow2 62.0%

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

      4. times-frac 68.3%

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

   10. Simplified 68.3%

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

   if 5.00000000000000018e-11 < k

   1. Initial program 42.6%

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

      1. associate-/r* 42.6%

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

      2. associate-*l* 42.6%

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

      3. sqr-neg 42.6%

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

      4. associate-*l* 42.6%

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

      5. *-commutative 42.6%

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

      6. sqr-neg 42.6%

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

      7. associate-*l/ 42.6%

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

      8. associate-*r/ 42.6%

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

      9. associate-/r/ 42.6%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in k around inf 64.1%

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

      1. times-frac 60.1%

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

      2. unpow2 60.1%

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

      3. unpow2 60.1%

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

      4. times-frac 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in k around 0 51.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 51.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      2. fma-def 51.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      3. unpow2 51.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      4. associate-*r/ 52.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \frac{\ell}{t}}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      5. unpow2 52.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right)\right) \]

      6. associate-/l* 58.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot t}{\ell}}}\right)\right) \]

      7. unpow2 58.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell}}\right)\right) \]

   9. Simplified 58.2%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \frac{\ell}{t}, \frac{\ell}{\frac{\left(k \cdot k\right) \cdot t}{\ell}}\right)}\right) \]

   10. Taylor expanded in k around inf 53.5%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. times-frac 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]

       2. unpow2 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       3. unpow2 50.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

       4. associate-*r/ 56.9%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right)\right) \]

   12. Simplified 56.9%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 16: 63.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.8e-14)
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.8e-14) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.8d-14) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.8e-14) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.8e-14:
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.8e-14)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.8e-14)
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.8e-14], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.8000000000000001e-14

   1. Initial program 61.7%

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

      1. associate-/r* 61.7%

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

      2. associate-*l* 53.1%

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

      3. sqr-neg 53.1%

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

      4. associate-*l* 61.7%

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

      5. *-commutative 61.7%

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

      6. sqr-neg 61.7%

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

      7. associate-/r* 61.7%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 62.1%

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

      2. associate-/r/ 61.7%

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

      3. add-cube-cbrt 61.7%

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

   5. Applied egg-rr 74.5%

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

      1. pow-plus 74.5%

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

      2. metadata-eval 74.5%

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

      3. associate-/r/ 74.6%

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

      4. associate-/r/ 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in k around 0 53.9%

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

      1. unpow2 53.9%

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

      2. associate-*r* 62.0%

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

      3. unpow2 62.0%

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

      4. times-frac 68.3%

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

   10. Simplified 68.3%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if 2.8000000000000001e-14 < k

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 64.1%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 60.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 60.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 60.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. times-frac 69.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}\right) \]

   6. Simplified 69.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)\right)} \]

   7. Taylor expanded in k around 0 49.1%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. *-commutative 49.1%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      2. associate-/r* 47.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

      3. unpow2 47.6%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{4}} \]

      4. associate-*r/ 54.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{{k}^{4}} \]

   9. Simplified 54.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 17: 57.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.4%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 56.7%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 56.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 56.1%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 56.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 62.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. times-frac 61.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

   2. unpow2 61.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

   3. unpow2 61.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

   4. times-frac 67.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)}\right) \]

6. Simplified 67.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{t}\right)\right)} \]

7. Taylor expanded in k around 0 52.7%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 52.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 52.7%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

9. Simplified 52.7%

   \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

10. Taylor expanded in l around 0 52.7%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 52.7%

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

    2. *-commutative 52.7%

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

    3. times-frac 56.4%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

12. Simplified 56.4%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

13. Final simplification 56.4%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Reproduce

herbie shell --seed 2023275 
(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))))