Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 89.3%

Time: 36.6s

Alternatives: 22

Speedup: 3.8×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-8} \lor \neg \left(t \leq 9.5 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1e-8) (not (<= t 9.5e-87)))
   (/
    (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 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1e-8) || !(t <= 9.5e-87)) {
		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 * (((cos(k) * (l / t)) / (k * k)) * (l / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1e-8) || !(t <= 9.5e-87)) {
		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.cos(k) * (l / t)) / (k * k)) * (l / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1e-8) || !(t <= 9.5e-87))
		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(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1e-8], N[Not[LessEqual[t, 9.5e-87]], $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[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1e-8 or 9.5e-87 < t

   1. Initial program 64.4%

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

      1. associate-/r* 64.4%

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

      2. associate-*l* 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-*l* 64.4%

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

      5. *-commutative 64.4%

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

      6. sqr-neg 64.4%

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

      7. associate-/r* 64.4%

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

   3. Simplified 64.9%

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

      1. associate-/l/ 64.9%

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

      2. associate-/r/ 64.4%

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

      3. add-cube-cbrt 64.3%

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

   5. Applied egg-rr 77.3%

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

      1. pow-plus 77.2%

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

      2. metadata-eval 77.2%

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

      3. associate-/r/ 77.3%

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

      4. associate-/r/ 77.3%

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

   7. Simplified 77.3%

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

      1. cbrt-prod 88.5%

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

   9. Applied egg-rr 88.5%

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

       1. cbrt-div 96.8%

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

   11. Applied egg-rr 96.8%

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

   if -1e-8 < t < 9.5e-87

   1. Initial program 45.3%

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

      1. associate-/r* 45.3%

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

      2. associate-*l* 45.3%

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

      3. sqr-neg 45.3%

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

      4. associate-*l* 45.3%

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

      5. *-commutative 45.3%

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

      6. sqr-neg 45.3%

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

      7. associate-/r* 45.3%

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

   3. Simplified 45.3%

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

      1. associate-/l/ 45.3%

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

      2. associate-/r/ 45.3%

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

      3. add-cube-cbrt 45.3%

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

   5. Applied egg-rr 63.2%

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

      1. pow-plus 63.2%

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

      2. metadata-eval 63.2%

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

      3. associate-/r/ 63.2%

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

      4. associate-/r/ 63.2%

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

   7. Simplified 63.2%

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

      1. cbrt-prod 63.0%

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

   9. Applied egg-rr 63.0%

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

       1. cbrt-div 63.0%

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

   11. Applied egg-rr 63.0%

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

   12. Taylor expanded in k around inf 76.2%

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

       1. *-commutative 76.2%

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

       2. times-frac 75.3%

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

       3. unpow2 75.3%

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

       4. unpow2 75.3%

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

       5. times-frac 84.6%

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

       6. associate-*r* 87.1%

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

       7. unpow2 87.1%

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

       8. associate-*l/ 87.2%

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

       9. unpow2 87.2%

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

   14. Simplified 87.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-8} \lor \neg \left(t \leq 9.5 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(t_1 + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ (+ t_1 1.0) 1.0)))
        1e+226)
     (/
      (* (* l l) (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (tan k))) 3.0)))
      (+ 2.0 t_1))
     (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = ((l * l) * (2.0 / pow(((t * cbrt(sin(k))) * cbrt(tan(k))), 3.0))) / (2.0 + t_1);
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / 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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = ((l * l) * (2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt(Math.tan(k))), 3.0))) / (2.0 + t_1);
	} else {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / 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(Float64(t_1 + 1.0) + 1.0))) <= 1e+226)
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(tan(k))) ^ 3.0))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (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[(N[(t$95$1 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+226], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999961e225

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. associate-*l* 71.9%

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

      3. sqr-neg 71.9%

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

      4. associate-*l* 79.4%

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

      5. *-commutative 79.4%

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

      6. sqr-neg 79.4%

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

      7. associate-*l/ 80.0%

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

      8. associate-*r/ 78.7%

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

      9. associate-/r/ 79.1%

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

   3. Simplified 79.1%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 78.9%

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

      3. *-commutative 78.9%

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

      4. cbrt-prod 78.9%

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

      5. cbrt-prod 78.7%

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

      6. rem-cbrt-cube 88.0%

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

   5. Applied egg-rr 88.0%

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

   if 9.99999999999999961e225 < (/.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 23.4%

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

      1. associate-/r* 23.4%

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

      2. associate-*l* 23.4%

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

      3. sqr-neg 23.4%

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

      4. associate-*l* 23.4%

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

      5. *-commutative 23.4%

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

      6. sqr-neg 23.4%

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

      7. associate-/r* 23.4%

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

   3. Simplified 23.4%

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

      1. associate-/l/ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. add-cube-cbrt 23.4%

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

   5. Applied egg-rr 46.6%

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

      1. pow-plus 46.6%

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

      2. metadata-eval 46.6%

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

      3. associate-/r/ 46.6%

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

      4. associate-/r/ 46.6%

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

   7. Simplified 46.6%

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

      1. cbrt-prod 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. cbrt-div 62.3%

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

   11. Applied egg-rr 62.3%

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

   12. Taylor expanded in k around inf 60.1%

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

       1. *-commutative 60.1%

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

       2. times-frac 59.2%

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

       3. unpow2 59.2%

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

       4. unpow2 59.2%

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

       5. times-frac 72.0%

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

       6. associate-*r* 74.8%

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

       7. unpow2 74.8%

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

       8. associate-*l/ 74.9%

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

       9. unpow2 74.9%

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

   14. Simplified 74.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(t_1 + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ (+ t_1 1.0) 1.0)))
        1e+226)
     (/
      (pow (* (/ (cbrt (/ 2.0 (tan k))) t) (cbrt (* l (/ l (sin k))))) 3.0)
      (+ 2.0 t_1))
     (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * cbrt((l * (l / sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / 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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		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.cos(k) * (l / t)) / (k * k)) * (l / 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(Float64(t_1 + 1.0) + 1.0))) <= 1e+226)
		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(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (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[(N[(t$95$1 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+226], 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[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999961e225

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. associate-*l* 71.9%

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

      3. sqr-neg 71.9%

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

      4. associate-*l* 79.4%

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

      5. *-commutative 79.4%

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

      6. sqr-neg 79.4%

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

      7. associate-/r* 79.4%

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

   3. Simplified 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-/r/ 79.4%

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

      3. add-cube-cbrt 79.3%

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

   5. Applied egg-rr 88.6%

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

      1. pow-plus 88.6%

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

      2. metadata-eval 88.6%

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

      3. associate-/r/ 88.6%

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

      4. associate-/r/ 88.7%

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

   7. Simplified 88.7%

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

   if 9.99999999999999961e225 < (/.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 23.4%

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

      1. associate-/r* 23.4%

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

      2. associate-*l* 23.4%

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

      3. sqr-neg 23.4%

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

      4. associate-*l* 23.4%

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

      5. *-commutative 23.4%

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

      6. sqr-neg 23.4%

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

      7. associate-/r* 23.4%

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

   3. Simplified 23.4%

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

      1. associate-/l/ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. add-cube-cbrt 23.4%

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

   5. Applied egg-rr 46.6%

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

      1. pow-plus 46.6%

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

      2. metadata-eval 46.6%

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

      3. associate-/r/ 46.6%

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

      4. associate-/r/ 46.6%

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

   7. Simplified 46.6%

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

      1. cbrt-prod 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. cbrt-div 62.3%

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

   11. Applied egg-rr 62.3%

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

   12. Taylor expanded in k around inf 60.1%

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

       1. *-commutative 60.1%

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

       2. times-frac 59.2%

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

       3. unpow2 59.2%

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

       4. unpow2 59.2%

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

       5. times-frac 72.0%

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

       6. associate-*r* 74.8%

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

       7. unpow2 74.8%

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

       8. associate-*l/ 74.9%

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

       9. unpow2 74.9%

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

   14. Simplified 74.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(t_1 + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3} \cdot \frac{\ell \cdot \frac{\ell}{\sin k}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ (+ t_1 1.0) 1.0)))
        1e+226)
     (*
      (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0)
      (/ (* l (/ l (sin k))) (+ 2.0 t_1)))
     (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = pow((cbrt((2.0 / tan(k))) / t), 3.0) * ((l * (l / sin(k))) / (2.0 + t_1));
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / 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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0) * ((l * (l / Math.sin(k))) / (2.0 + t_1));
	} else {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / 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(Float64(t_1 + 1.0) + 1.0))) <= 1e+226)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0) * Float64(Float64(l * Float64(l / sin(k))) / Float64(2.0 + t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (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[(N[(t$95$1 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+226], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999961e225

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. associate-*l* 71.9%

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

      3. sqr-neg 71.9%

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

      4. associate-*l* 79.4%

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

      5. *-commutative 79.4%

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

      6. sqr-neg 79.4%

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

      7. associate-/r* 79.4%

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

   3. Simplified 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-/r/ 79.4%

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

      3. add-cube-cbrt 79.3%

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

   5. Applied egg-rr 88.6%

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

      1. pow-plus 88.6%

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

      2. metadata-eval 88.6%

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

      3. associate-/r/ 88.6%

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

      4. associate-/r/ 88.7%

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

   7. Simplified 88.7%

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

      1. div-inv 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*l/ 86.3%

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

   9. Applied egg-rr 86.3%

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

       1. associate-*r/ 86.3%

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

       2. *-rgt-identity 86.3%

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

       3. cube-prod 84.3%

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

       4. unpow2 84.3%

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

       5. rem-cube-cbrt 84.3%

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

       6. associate-/l* 83.7%

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

       7. unpow2 83.7%

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

       8. associate-/l* 85.5%

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

       9. associate-/r/ 85.5%

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

   11. Simplified 85.5%

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

       1. *-un-lft-identity 85.5%

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

       2. associate-/r/ 85.3%

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

       3. *-commutative 85.3%

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

   13. Applied egg-rr 85.3%

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

       1. *-lft-identity 85.3%

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

   15. Simplified 85.3%

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

   if 9.99999999999999961e225 < (/.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 23.4%

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

      1. associate-/r* 23.4%

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

      2. associate-*l* 23.4%

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

      3. sqr-neg 23.4%

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

      4. associate-*l* 23.4%

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

      5. *-commutative 23.4%

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

      6. sqr-neg 23.4%

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

      7. associate-/r* 23.4%

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

   3. Simplified 23.4%

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

      1. associate-/l/ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. add-cube-cbrt 23.4%

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

   5. Applied egg-rr 46.6%

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

      1. pow-plus 46.6%

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

      2. metadata-eval 46.6%

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

      3. associate-/r/ 46.6%

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

      4. associate-/r/ 46.6%

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

   7. Simplified 46.6%

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

      1. cbrt-prod 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. cbrt-div 62.3%

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

   11. Applied egg-rr 62.3%

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

   12. Taylor expanded in k around inf 60.1%

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

       1. *-commutative 60.1%

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

       2. times-frac 59.2%

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

       3. unpow2 59.2%

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

       4. unpow2 59.2%

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

       5. times-frac 72.0%

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

       6. associate-*r* 74.8%

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

       7. unpow2 74.8%

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

       8. associate-*l/ 74.9%

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

       9. unpow2 74.9%

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

   14. Simplified 74.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3} \cdot \frac{\ell \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(t_1 + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k}}{\frac{2 + t_1}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ (+ t_1 1.0) 1.0)))
        1e+226)
     (/
      (* l (/ l (sin k)))
      (/ (+ 2.0 t_1) (pow (/ (cbrt (/ 2.0 (tan k))) t) 3.0)))
     (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = (l * (l / sin(k))) / ((2.0 + t_1) / pow((cbrt((2.0 / tan(k))) / t), 3.0));
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / 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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = (l * (l / Math.sin(k))) / ((2.0 + t_1) / Math.pow((Math.cbrt((2.0 / Math.tan(k))) / t), 3.0));
	} else {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / 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(Float64(t_1 + 1.0) + 1.0))) <= 1e+226)
		tmp = Float64(Float64(l * Float64(l / sin(k))) / Float64(Float64(2.0 + t_1) / (Float64(cbrt(Float64(2.0 / tan(k))) / t) ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (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[(N[(t$95$1 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+226], N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + t$95$1), $MachinePrecision] / N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999961e225

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. associate-*l* 71.9%

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

      3. sqr-neg 71.9%

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

      4. associate-*l* 79.4%

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

      5. *-commutative 79.4%

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

      6. sqr-neg 79.4%

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

      7. associate-/r* 79.4%

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

   3. Simplified 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-/r/ 79.4%

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

      3. add-cube-cbrt 79.3%

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

   5. Applied egg-rr 88.6%

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

      1. pow-plus 88.6%

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

      2. metadata-eval 88.6%

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

      3. associate-/r/ 88.6%

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

      4. associate-/r/ 88.7%

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

   7. Simplified 88.7%

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

      1. div-inv 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*l/ 86.3%

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

   9. Applied egg-rr 86.3%

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

       1. associate-*r/ 86.3%

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

       2. *-rgt-identity 86.3%

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

       3. cube-prod 84.3%

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

       4. unpow2 84.3%

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

       5. rem-cube-cbrt 84.3%

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

       6. associate-/l* 83.7%

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

       7. unpow2 83.7%

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

       8. associate-/l* 85.5%

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

       9. associate-/r/ 85.5%

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

   11. Simplified 85.5%

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

   if 9.99999999999999961e225 < (/.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 23.4%

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

      1. associate-/r* 23.4%

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

      2. associate-*l* 23.4%

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

      3. sqr-neg 23.4%

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

      4. associate-*l* 23.4%

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

      5. *-commutative 23.4%

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

      6. sqr-neg 23.4%

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

      7. associate-/r* 23.4%

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

   3. Simplified 23.4%

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

      1. associate-/l/ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. add-cube-cbrt 23.4%

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

   5. Applied egg-rr 46.6%

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

      1. pow-plus 46.6%

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

      2. metadata-eval 46.6%

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

      3. associate-/r/ 46.6%

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

      4. associate-/r/ 46.6%

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

   7. Simplified 46.6%

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

      1. cbrt-prod 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. cbrt-div 62.3%

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

   11. Applied egg-rr 62.3%

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

   12. Taylor expanded in k around inf 60.1%

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

       1. *-commutative 60.1%

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

       2. times-frac 59.2%

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

       3. unpow2 59.2%

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

       4. unpow2 59.2%

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

       5. times-frac 72.0%

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

       6. associate-*r* 74.8%

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

       7. unpow2 74.8%

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

       8. associate-*l/ 74.9%

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

       9. unpow2 74.9%

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

   14. Simplified 74.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k}}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(t_1 + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{2 + t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
          (+ (+ t_1 1.0) 1.0)))
        1e+226)
     (* (/ l (sin k)) (* l (/ (/ (/ 2.0 (tan k)) (pow t 3.0)) (+ 2.0 t_1))))
     (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = (l / sin(k)) * (l * (((2.0 / tan(k)) / pow(t, 3.0)) / (2.0 + t_1)));
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / 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)))) * ((t_1 + 1.0d0) + 1.0d0))) <= 1d+226) then
        tmp = (l / sin(k)) * (l * (((2.0d0 / tan(k)) / (t ** 3.0d0)) / (2.0d0 + t_1)))
    else
        tmp = 2.0d0 * (((cos(k) * (l / t)) / (k * k)) * (l / (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226) {
		tmp = (l / Math.sin(k)) * (l * (((2.0 / Math.tan(k)) / Math.pow(t, 3.0)) / (2.0 + t_1)));
	} else {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / 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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226:
		tmp = (l / math.sin(k)) * (l * (((2.0 / math.tan(k)) / math.pow(t, 3.0)) / (2.0 + t_1)))
	else:
		tmp = 2.0 * (((math.cos(k) * (l / t)) / (k * k)) * (l / 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(Float64(t_1 + 1.0) + 1.0))) <= 1e+226)
		tmp = Float64(Float64(l / sin(k)) * Float64(l * Float64(Float64(Float64(2.0 / tan(k)) / (t ^ 3.0)) / Float64(2.0 + t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (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)))) * ((t_1 + 1.0) + 1.0))) <= 1e+226)
		tmp = (l / sin(k)) * (l * (((2.0 / tan(k)) / (t ^ 3.0)) / (2.0 + t_1)));
	else
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / (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[(N[(t$95$1 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+226], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999961e225

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. associate-*l* 71.9%

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

      3. sqr-neg 71.9%

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

      4. associate-*l* 79.4%

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

      5. *-commutative 79.4%

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

      6. sqr-neg 79.4%

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

      7. associate-/r* 79.4%

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

   3. Simplified 79.9%

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

      1. associate-/l/ 79.9%

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

      2. associate-/r/ 79.4%

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

      3. add-cube-cbrt 79.3%

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

   5. Applied egg-rr 88.6%

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

      1. pow-plus 88.6%

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

      2. metadata-eval 88.6%

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

      3. associate-/r/ 88.6%

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

      4. associate-/r/ 88.7%

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

   7. Simplified 88.7%

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

      1. div-inv 88.7%

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

      2. *-commutative 88.7%

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

      3. associate-*l/ 86.3%

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

   9. Applied egg-rr 86.3%

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

       1. associate-*r/ 86.3%

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

       2. *-rgt-identity 86.3%

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

       3. cube-prod 84.3%

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

       4. unpow2 84.3%

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

       5. rem-cube-cbrt 84.3%

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

       6. associate-/l* 83.7%

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

       7. unpow2 83.7%

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

       8. associate-/l* 85.5%

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

       9. associate-/r/ 85.5%

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

   11. Simplified 85.5%

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

       1. div-inv 85.5%

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

       2. *-commutative 85.5%

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

   13. Applied egg-rr 85.5%

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

       1. *-commutative 85.5%

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

       2. associate-*l* 86.5%

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

       3. associate-/r/ 86.4%

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

       4. associate-*l/ 86.5%

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

       5. metadata-eval 86.5%

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

       6. cube-prod 86.5%

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

       7. *-lft-identity 86.5%

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

       8. cube-div 83.8%

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

       9. rem-cube-cbrt 84.0%

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

   15. Simplified 84.0%

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

   if 9.99999999999999961e225 < (/.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 23.4%

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

      1. associate-/r* 23.4%

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

      2. associate-*l* 23.4%

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

      3. sqr-neg 23.4%

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

      4. associate-*l* 23.4%

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

      5. *-commutative 23.4%

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

      6. sqr-neg 23.4%

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

      7. associate-/r* 23.4%

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

   3. Simplified 23.4%

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

      1. associate-/l/ 23.4%

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

      2. associate-/r/ 23.4%

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

      3. add-cube-cbrt 23.4%

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

   5. Applied egg-rr 46.6%

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

      1. pow-plus 46.6%

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

      2. metadata-eval 46.6%

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

      3. associate-/r/ 46.6%

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

      4. associate-/r/ 46.6%

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

   7. Simplified 46.6%

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

      1. cbrt-prod 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. cbrt-div 62.3%

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

   11. Applied egg-rr 62.3%

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

   12. Taylor expanded in k around inf 60.1%

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

       1. *-commutative 60.1%

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

       2. times-frac 59.2%

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

       3. unpow2 59.2%

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

       4. unpow2 59.2%

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

       5. times-frac 72.0%

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

       6. associate-*r* 74.8%

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

       7. unpow2 74.8%

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

       8. associate-*l/ 74.9%

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

       9. unpow2 74.9%

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

   14. Simplified 74.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 10^{+226}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 7: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-7} \lor \neg \left(t \leq 5.8 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{{\left(\sqrt[3]{\ell} \cdot \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.3e-7) (not (<= t 5.8e-87)))
   (/
    (pow
     (* (cbrt l) (* (/ (cbrt (/ 2.0 (tan k))) t) (cbrt (/ l (sin k)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-7) || !(t <= 5.8e-87)) {
		tmp = pow((cbrt(l) * ((cbrt((2.0 / tan(k))) / t) * cbrt((l / sin(k))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-7) || !(t <= 5.8e-87)) {
		tmp = Math.pow((Math.cbrt(l) * ((Math.cbrt((2.0 / Math.tan(k))) / t) * Math.cbrt((l / Math.sin(k))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.3e-7) || !(t <= 5.8e-87))
		tmp = Float64((Float64(cbrt(l) * Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * cbrt(Float64(l / sin(k))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.3e-7], N[Not[LessEqual[t, 5.8e-87]], $MachinePrecision]], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $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[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.3000000000000002e-7 or 5.7999999999999998e-87 < t

   1. Initial program 64.4%

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

      1. associate-/r* 64.4%

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

      2. associate-*l* 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-*l* 64.4%

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

      5. *-commutative 64.4%

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

      6. sqr-neg 64.4%

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

      7. associate-/r* 64.4%

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

   3. Simplified 64.9%

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

      1. associate-/l/ 64.9%

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

      2. associate-/r/ 64.4%

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

      3. add-cube-cbrt 64.3%

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

   5. Applied egg-rr 77.3%

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

      1. pow-plus 77.2%

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

      2. metadata-eval 77.2%

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

      3. associate-/r/ 77.3%

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

      4. associate-/r/ 77.3%

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

   7. Simplified 77.3%

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

      1. cbrt-prod 88.5%

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

   9. Applied egg-rr 88.5%

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

       1. cbrt-div 96.8%

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

   11. Applied egg-rr 96.8%

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

       1. associate-*l/ 95.5%

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

       2. *-commutative 95.5%

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

       3. cbrt-undiv 88.5%

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

   13. Applied egg-rr 88.5%

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

       1. associate-*l/ 88.5%

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

       2. *-commutative 88.5%

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

       3. associate-*l* 88.5%

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

   15. Simplified 88.5%

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

   if -3.3000000000000002e-7 < t < 5.7999999999999998e-87

   1. Initial program 45.3%

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

      1. associate-/r* 45.3%

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

      2. associate-*l* 45.3%

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

      3. sqr-neg 45.3%

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

      4. associate-*l* 45.3%

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

      5. *-commutative 45.3%

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

      6. sqr-neg 45.3%

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

      7. associate-/r* 45.3%

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

   3. Simplified 45.3%

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

      1. associate-/l/ 45.3%

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

      2. associate-/r/ 45.3%

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

      3. add-cube-cbrt 45.3%

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

   5. Applied egg-rr 63.2%

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

      1. pow-plus 63.2%

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

      2. metadata-eval 63.2%

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

      3. associate-/r/ 63.2%

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

      4. associate-/r/ 63.2%

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

   7. Simplified 63.2%

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

      1. cbrt-prod 63.0%

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

   9. Applied egg-rr 63.0%

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

       1. cbrt-div 63.0%

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

   11. Applied egg-rr 63.0%

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

   12. Taylor expanded in k around inf 76.2%

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

       1. *-commutative 76.2%

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

       2. times-frac 75.3%

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

       3. unpow2 75.3%

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

       4. unpow2 75.3%

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

       5. times-frac 84.6%

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

       6. associate-*r* 87.1%

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

       7. unpow2 87.1%

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

       8. associate-*l/ 87.2%

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

       9. unpow2 87.2%

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

   14. Simplified 87.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-7} \lor \neg \left(t \leq 5.8 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{{\left(\sqrt[3]{\ell} \cdot \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 8: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\frac{\ell}{\sin k}}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{\sqrt[3]{\frac{2}{\tan k}}}{t}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\ell} \cdot \left(t_3 \cdot t_1\right)\right)}^{3}}{t_2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t_3 \cdot \left(\sqrt[3]{\ell} \cdot t_1\right)\right)}^{3}}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (/ l (sin k))))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (/ (cbrt (/ 2.0 (tan k))) t)))
   (if (<= t -2.2e-7)
     (/ (pow (* (cbrt l) (* t_3 t_1)) 3.0) t_2)
     (if (<= t 2.35e-89)
       (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (pow (sin k) 2.0))))
       (/ (pow (* t_3 (* (cbrt l) t_1)) 3.0) t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((l / sin(k)));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = cbrt((2.0 / tan(k))) / t;
	double tmp;
	if (t <= -2.2e-7) {
		tmp = pow((cbrt(l) * (t_3 * t_1)), 3.0) / t_2;
	} else if (t <= 2.35e-89) {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / pow(sin(k), 2.0)));
	} else {
		tmp = pow((t_3 * (cbrt(l) * t_1)), 3.0) / t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((l / Math.sin(k)));
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = Math.cbrt((2.0 / Math.tan(k))) / t;
	double tmp;
	if (t <= -2.2e-7) {
		tmp = Math.pow((Math.cbrt(l) * (t_3 * t_1)), 3.0) / t_2;
	} else if (t <= 2.35e-89) {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((t_3 * (Math.cbrt(l) * t_1)), 3.0) / t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(l / sin(k)))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(cbrt(Float64(2.0 / tan(k))) / t)
	tmp = 0.0
	if (t <= -2.2e-7)
		tmp = Float64((Float64(cbrt(l) * Float64(t_3 * t_1)) ^ 3.0) / t_2);
	elseif (t <= 2.35e-89)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (sin(k) ^ 2.0))));
	else
		tmp = Float64((Float64(t_3 * Float64(cbrt(l) * t_1)) ^ 3.0) / t_2);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -2.2e-7], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t, 2.35e-89], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$3 * N[(N[Power[l, 1/3], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.2000000000000001e-7

   1. Initial program 62.0%

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

      1. associate-/r* 62.0%

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

      2. associate-*l* 53.5%

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

      3. sqr-neg 53.5%

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

      4. associate-*l* 62.0%

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

      5. *-commutative 62.0%

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

      6. sqr-neg 62.0%

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

      7. associate-/r* 62.0%

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

   3. Simplified 63.5%

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

      1. associate-/l/ 63.5%

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

      2. associate-/r/ 62.0%

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

      3. add-cube-cbrt 61.9%

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

   5. Applied egg-rr 77.0%

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

      1. pow-plus 77.0%

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

      2. metadata-eval 77.0%

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

      3. associate-/r/ 77.0%

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

      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. Step-by-step derivation

      1. cbrt-prod 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. cbrt-div 97.1%

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

   11. Applied egg-rr 97.1%

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

       1. associate-*l/ 95.6%

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

       2. *-commutative 95.6%

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

       3. cbrt-undiv 88.1%

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

   13. Applied egg-rr 88.1%

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

       1. associate-*l/ 88.0%

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

       2. *-commutative 88.0%

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

       3. associate-*l* 88.1%

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

   15. Simplified 88.1%

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

   if -2.2000000000000001e-7 < t < 2.34999999999999998e-89

   1. Initial program 45.3%

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

      1. associate-/r* 45.3%

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

      2. associate-*l* 45.3%

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

      3. sqr-neg 45.3%

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

      4. associate-*l* 45.3%

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

      5. *-commutative 45.3%

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

      6. sqr-neg 45.3%

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

      7. associate-/r* 45.3%

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

   3. Simplified 45.3%

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

      1. associate-/l/ 45.3%

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

      2. associate-/r/ 45.3%

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

      3. add-cube-cbrt 45.3%

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

   5. Applied egg-rr 63.2%

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

      1. pow-plus 63.2%

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

      2. metadata-eval 63.2%

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

      3. associate-/r/ 63.2%

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

      4. associate-/r/ 63.2%

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

   7. Simplified 63.2%

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

      1. cbrt-prod 63.0%

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

   9. Applied egg-rr 63.0%

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

       1. cbrt-div 63.0%

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

   11. Applied egg-rr 63.0%

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

   12. Taylor expanded in k around inf 76.2%

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

       1. *-commutative 76.2%

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

       2. times-frac 75.3%

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

       3. unpow2 75.3%

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

       4. unpow2 75.3%

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

       5. times-frac 84.6%

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

       6. associate-*r* 87.1%

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

       7. unpow2 87.1%

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

       8. associate-*l/ 87.2%

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

       9. unpow2 87.2%

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

   14. Simplified 87.2%

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

   if 2.34999999999999998e-89 < t

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. associate-*l* 59.0%

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

      3. sqr-neg 59.0%

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

      4. associate-*l* 66.2%

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

      5. *-commutative 66.2%

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

      6. sqr-neg 66.2%

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

      7. associate-/r* 66.2%

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

   3. Simplified 66.0%

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

      1. associate-/l/ 66.0%

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

      2. associate-/r/ 66.2%

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

      3. add-cube-cbrt 66.1%

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

   5. Applied egg-rr 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/ 77.5%

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

   7. Simplified 77.5%

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

      1. cbrt-prod 88.9%

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

   9. Applied egg-rr 88.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\ell} \cdot \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 9: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \sqrt[3]{\frac{2}{\tan k}}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{{\left(\frac{t_3 \cdot t_1}{t}\right)}^{3}}{t_2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t_3}{t} \cdot t_1\right)}^{3}}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cbrt l) (cbrt (/ l (sin k)))))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (cbrt (/ 2.0 (tan k)))))
   (if (<= t -6.5e-18)
     (/ (pow (/ (* t_3 t_1) t) 3.0) t_2)
     (if (<= t 2.35e-89)
       (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (pow (sin k) 2.0))))
       (/ (pow (* (/ t_3 t) t_1) 3.0) t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt(l) * cbrt((l / sin(k)));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = cbrt((2.0 / tan(k)));
	double tmp;
	if (t <= -6.5e-18) {
		tmp = pow(((t_3 * t_1) / t), 3.0) / t_2;
	} else if (t <= 2.35e-89) {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / pow(sin(k), 2.0)));
	} else {
		tmp = pow(((t_3 / t) * t_1), 3.0) / t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt(l) * Math.cbrt((l / Math.sin(k)));
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = Math.cbrt((2.0 / Math.tan(k)));
	double tmp;
	if (t <= -6.5e-18) {
		tmp = Math.pow(((t_3 * t_1) / t), 3.0) / t_2;
	} else if (t <= 2.35e-89) {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow(((t_3 / t) * t_1), 3.0) / t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(cbrt(l) * cbrt(Float64(l / sin(k))))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = cbrt(Float64(2.0 / tan(k)))
	tmp = 0.0
	if (t <= -6.5e-18)
		tmp = Float64((Float64(Float64(t_3 * t_1) / t) ^ 3.0) / t_2);
	elseif (t <= 2.35e-89)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (sin(k) ^ 2.0))));
	else
		tmp = Float64((Float64(Float64(t_3 / t) * t_1) ^ 3.0) / t_2);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -6.5e-18], N[(N[Power[N[(N[(t$95$3 * t$95$1), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t, 2.35e-89], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(t$95$3 / t), $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.50000000000000008e-18

   1. Initial program 61.5%

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

      1. associate-/r* 61.5%

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

      2. associate-*l* 53.4%

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

      3. sqr-neg 53.4%

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

      4. associate-*l* 61.5%

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

      5. *-commutative 61.5%

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

      6. sqr-neg 61.5%

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

      7. associate-/r* 61.4%

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

   3. Simplified 62.9%

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

      1. associate-/l/ 62.9%

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

      2. associate-/r/ 61.5%

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

      3. add-cube-cbrt 61.4%

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

   5. Applied egg-rr 75.6%

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

      1. pow-plus 75.6%

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

      2. metadata-eval 75.6%

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

      3. associate-/r/ 75.6%

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

      4. associate-/r/ 75.7%

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

   7. Simplified 75.7%

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

      1. cbrt-prod 85.9%

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

   9. Applied egg-rr 85.9%

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

       1. cbrt-div 94.4%

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

   11. Applied egg-rr 94.4%

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

       1. associate-*l/ 93.0%

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

       2. *-commutative 93.0%

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

       3. cbrt-undiv 86.0%

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

   13. Applied egg-rr 86.0%

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

   if -6.50000000000000008e-18 < t < 2.34999999999999998e-89

   1. Initial program 45.0%

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

      1. associate-/r* 45.0%

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

      2. associate-*l* 45.0%

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

      3. sqr-neg 45.0%

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

      4. associate-*l* 45.0%

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

      5. *-commutative 45.0%

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

      6. sqr-neg 45.0%

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

      7. associate-/r* 45.0%

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

   3. Simplified 45.0%

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

      1. associate-/l/ 45.0%

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

      2. associate-/r/ 45.0%

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

      3. add-cube-cbrt 45.0%

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

   5. Applied egg-rr 63.5%

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

      1. pow-plus 63.5%

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

      2. metadata-eval 63.5%

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

      3. associate-/r/ 63.5%

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

      4. associate-/r/ 63.5%

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

   7. Simplified 63.5%

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

      1. cbrt-prod 63.3%

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

   9. Applied egg-rr 63.3%

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

       1. cbrt-div 63.3%

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

   11. Applied egg-rr 63.3%

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

   12. Taylor expanded in k around inf 77.9%

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

       1. *-commutative 77.9%

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

       2. times-frac 76.9%

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

       3. unpow2 76.9%

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

       4. unpow2 76.9%

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

       5. times-frac 86.6%

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

       6. associate-*r* 88.4%

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

       7. unpow2 88.4%

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

       8. associate-*l/ 88.4%

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

       9. unpow2 88.4%

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

   14. Simplified 88.4%

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

   if 2.34999999999999998e-89 < t

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. associate-*l* 59.0%

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

      3. sqr-neg 59.0%

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

      4. associate-*l* 66.2%

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

      5. *-commutative 66.2%

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

      6. sqr-neg 66.2%

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

      7. associate-/r* 66.2%

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

   3. Simplified 66.0%

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

      1. associate-/l/ 66.0%

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

      2. associate-/r/ 66.2%

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

      3. add-cube-cbrt 66.1%

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

   5. Applied egg-rr 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/ 77.5%

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

   7. Simplified 77.5%

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

      1. cbrt-prod 88.9%

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

   9. Applied egg-rr 88.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)}{t}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 10: 63.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2e-15)
   (/ (* l (/ l (sin k))) (* k (pow t 3.0)))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l t) (/ l (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e-15) {
		tmp = (l * (l / sin(k))) / (k * pow(t, 3.0));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2d-15) then
        tmp = (l * (l / sin(k))) / (k * (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * (l / (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e-15) {
		tmp = (l * (l / Math.sin(k))) / (k * Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * (l / Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2e-15:
		tmp = (l * (l / math.sin(k))) / (k * math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * (l / math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2e-15)
		tmp = Float64(Float64(l * Float64(l / sin(k))) / Float64(k * (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2e-15)
		tmp = (l * (l / sin(k))) / (k * (t ^ 3.0));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2e-15], N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t, 3.0], $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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.0000000000000002e-15

   1. Initial program 61.9%

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

      1. associate-/r* 61.9%

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

      2. associate-*l* 55.8%

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

      3. sqr-neg 55.8%

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

      4. associate-*l* 61.9%

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

      5. *-commutative 61.9%

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

      6. sqr-neg 61.9%

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

      7. associate-/r* 61.9%

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

   3. Simplified 62.3%

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

      1. associate-/l/ 62.3%

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

      2. associate-/r/ 61.9%

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

      3. add-cube-cbrt 61.8%

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

   5. Applied egg-rr 76.4%

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

      1. pow-plus 76.3%

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

      2. metadata-eval 76.3%

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

      3. associate-/r/ 76.4%

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

      4. associate-/r/ 76.4%

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

   7. Simplified 76.4%

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

      1. div-inv 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*l/ 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. associate-*r/ 71.8%

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

       2. *-rgt-identity 71.8%

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

       3. cube-prod 68.1%

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

       4. unpow2 68.1%

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

       5. rem-cube-cbrt 68.1%

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

       6. associate-/l* 67.6%

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

       7. unpow2 67.6%

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

       8. associate-/l* 71.8%

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

       9. associate-/r/ 71.8%

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

   11. Simplified 71.8%

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

   12. Taylor expanded in k around 0 64.2%

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

   if 2.0000000000000002e-15 < k

   1. Initial program 41.2%

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

      1. associate-/r* 41.2%

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

      2. associate-*l* 41.2%

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

      3. sqr-neg 41.2%

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

      4. associate-*l* 41.2%

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

      5. *-commutative 41.2%

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

      6. sqr-neg 41.2%

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

      7. associate-*l/ 41.2%

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

      8. associate-*r/ 41.2%

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

      9. associate-/r/ 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in k around inf 78.7%

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

      1. times-frac 78.7%

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

      2. unpow2 78.7%

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

      3. unpow2 78.7%

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

      4. *-commutative 78.7%

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

      5. times-frac 81.5%

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

   6. Simplified 81.5%

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

4. Final simplification 69.2%

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

Alternative 11: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-13)
   (/ (* l (/ l (sin k))) (* k (pow t 3.0)))
   (* 2.0 (* (/ (* (cos k) (/ l t)) (* k k)) (/ l (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-13) {
		tmp = (l * (l / sin(k))) / (k * pow(t, 3.0));
	} else {
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.5d-13) then
        tmp = (l * (l / sin(k))) / (k * (t ** 3.0d0))
    else
        tmp = 2.0d0 * (((cos(k) * (l / t)) / (k * k)) * (l / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-13) {
		tmp = (l * (l / Math.sin(k))) / (k * Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (((Math.cos(k) * (l / t)) / (k * k)) * (l / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.5e-13:
		tmp = (l * (l / math.sin(k))) / (k * math.pow(t, 3.0))
	else:
		tmp = 2.0 * (((math.cos(k) * (l / t)) / (k * k)) * (l / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-13)
		tmp = Float64(Float64(l * Float64(l / sin(k))) / Float64(k * (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l / t)) / Float64(k * k)) * Float64(l / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.5e-13)
		tmp = (l * (l / sin(k))) / (k * (t ^ 3.0));
	else
		tmp = 2.0 * (((cos(k) * (l / t)) / (k * k)) * (l / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.5e-13], N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.5000000000000001e-13

   1. Initial program 61.9%

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

      1. associate-/r* 61.9%

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

      2. associate-*l* 55.8%

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

      3. sqr-neg 55.8%

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

      4. associate-*l* 61.9%

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

      5. *-commutative 61.9%

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

      6. sqr-neg 61.9%

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

      7. associate-/r* 61.9%

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

   3. Simplified 62.3%

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

      1. associate-/l/ 62.3%

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

      2. associate-/r/ 61.9%

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

      3. add-cube-cbrt 61.8%

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

   5. Applied egg-rr 76.4%

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

      1. pow-plus 76.3%

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

      2. metadata-eval 76.3%

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

      3. associate-/r/ 76.4%

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

      4. associate-/r/ 76.4%

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

   7. Simplified 76.4%

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

      1. div-inv 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-*l/ 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. associate-*r/ 71.8%

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

       2. *-rgt-identity 71.8%

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

       3. cube-prod 68.1%

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

       4. unpow2 68.1%

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

       5. rem-cube-cbrt 68.1%

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

       6. associate-/l* 67.6%

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

       7. unpow2 67.6%

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

       8. associate-/l* 71.8%

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

       9. associate-/r/ 71.8%

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

   11. Simplified 71.8%

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

   12. Taylor expanded in k around 0 64.2%

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

   if 8.5000000000000001e-13 < k

   1. Initial program 41.2%

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

      1. associate-/r* 41.2%

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

      2. associate-*l* 41.2%

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

      3. sqr-neg 41.2%

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

      4. associate-*l* 41.2%

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

      5. *-commutative 41.2%

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

      6. sqr-neg 41.2%

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

      7. associate-/r* 41.2%

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

   3. Simplified 41.2%

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

      1. associate-/l/ 41.2%

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

      2. associate-/r/ 41.2%

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

      3. add-cube-cbrt 41.2%

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

   5. Applied egg-rr 57.7%

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

      1. pow-plus 57.7%

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

      2. metadata-eval 57.7%

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

      3. associate-/r/ 57.7%

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

      4. associate-/r/ 57.7%

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

   7. Simplified 57.7%

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

      1. cbrt-prod 65.7%

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

   9. Applied egg-rr 65.7%

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

       1. cbrt-div 65.6%

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

   11. Applied egg-rr 65.6%

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

   12. Taylor expanded in k around inf 78.7%

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

       1. *-commutative 78.7%

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

       2. times-frac 78.7%

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

       3. unpow2 78.7%

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

       4. unpow2 78.7%

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

       5. times-frac 81.5%

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

       6. associate-*r* 81.8%

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

       7. unpow2 81.8%

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

       8. associate-*l/ 81.9%

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

       9. unpow2 81.9%

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

   14. Simplified 81.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 12: 61.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 22500000:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{3}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 22500000.0)
   (/ (* l (/ l (sin k))) (* k (pow t 3.0)))
   (* 2.0 (/ (cos k) (* (/ 3.0 l) (/ (* t (* k k)) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 22500000.0) {
		tmp = (l * (l / sin(k))) / (k * pow(t, 3.0));
	} else {
		tmp = 2.0 * (cos(k) / ((3.0 / 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 <= 22500000.0d0) then
        tmp = (l * (l / sin(k))) / (k * (t ** 3.0d0))
    else
        tmp = 2.0d0 * (cos(k) / ((3.0d0 / 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 <= 22500000.0) {
		tmp = (l * (l / Math.sin(k))) / (k * Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (Math.cos(k) / ((3.0 / l) * ((t * (k * k)) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 22500000.0:
		tmp = (l * (l / math.sin(k))) / (k * math.pow(t, 3.0))
	else:
		tmp = 2.0 * (math.cos(k) / ((3.0 / l) * ((t * (k * k)) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 22500000.0)
		tmp = Float64(Float64(l * Float64(l / sin(k))) / Float64(k * (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(3.0 / 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 <= 22500000.0)
		tmp = (l * (l / sin(k))) / (k * (t ^ 3.0));
	else
		tmp = 2.0 * (cos(k) / ((3.0 / l) * ((t * (k * k)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 22500000.0], N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(3.0 / l), $MachinePrecision] * N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.25e7

   1. Initial program 61.5%

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

      1. associate-/r* 61.5%

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

      2. associate-*l* 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* 61.5%

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

      5. *-commutative 61.5%

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

      6. sqr-neg 61.5%

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

      7. associate-/r* 61.5%

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

   3. Simplified 61.9%

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

      1. associate-/l/ 61.9%

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

      2. associate-/r/ 61.5%

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

      3. add-cube-cbrt 61.4%

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

   5. Applied egg-rr 75.5%

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

      1. pow-plus 75.5%

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

      2. metadata-eval 75.5%

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

      3. associate-/r/ 75.5%

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

      4. associate-/r/ 75.5%

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

   7. Simplified 75.5%

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

      1. div-inv 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l/ 71.1%

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

   9. Applied egg-rr 71.1%

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

       1. associate-*r/ 71.1%

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

       2. *-rgt-identity 71.1%

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

       3. cube-prod 67.5%

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

       4. unpow2 67.5%

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

       5. rem-cube-cbrt 67.5%

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

       6. associate-/l* 67.0%

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

       7. unpow2 67.0%

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

       8. associate-/l* 71.1%

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

       9. associate-/r/ 71.1%

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

   11. Simplified 71.1%

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

   12. Taylor expanded in k around 0 64.3%

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

   if 2.25e7 < k

   1. Initial program 40.4%

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

      1. associate-/r* 40.4%

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

      2. associate-*l* 40.4%

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

      3. sqr-neg 40.4%

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

      4. associate-*l* 40.4%

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

      5. *-commutative 40.4%

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

      6. sqr-neg 40.4%

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

      7. associate-*l/ 40.4%

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

      8. associate-*r/ 40.4%

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

      9. associate-/r/ 40.4%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in k around inf 78.2%

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

      1. times-frac 78.3%

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

      2. unpow2 78.3%

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

      3. unpow2 78.3%

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

      4. *-commutative 78.3%

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

   6. Simplified 78.3%

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

   7. Taylor expanded in k around 0 69.5%

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

      1. +-commutative 69.5%

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

      2. fma-def 69.5%

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

      3. unpow2 69.5%

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

      4. associate-/l* 69.5%

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

      5. *-commutative 69.5%

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

      6. associate-/r* 69.6%

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

      7. unpow2 69.6%

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

      8. associate-/l* 72.7%

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

      9. unpow2 72.7%

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

   9. Simplified 72.7%

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

   10. Taylor expanded in l around 0 69.6%

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

       1. associate-/l* 69.6%

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

       2. unpow2 69.6%

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

       3. *-commutative 69.6%

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

       4. unpow2 69.6%

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

       5. associate-*l* 72.7%

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

       6. associate-*r/ 72.7%

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

       7. metadata-eval 72.7%

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

       8. *-commutative 72.7%

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

       9. associate-/r* 72.7%

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

       10. unpow2 72.7%

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

   12. Simplified 72.7%

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

   13. Taylor expanded in k around inf 69.5%

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

       1. associate-*r/ 69.5%

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

       2. unpow2 69.5%

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

       3. times-frac 74.5%

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

       4. unpow2 74.5%

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

   15. Simplified 74.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 22500000:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\sin k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{3}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \]

Alternative 13: 68.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{k \cdot k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -4.9e-17)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 4.6e-90)
     (*
      2.0
      (/
       (cos k)
       (*
        (/ k l)
        (/ k (* l (+ (/ 0.3333333333333333 t) (/ (/ 1.0 t) (* k k))))))))
     (/ (* (/ l k) (/ l k)) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.9e-17) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 4.6e-90) {
		tmp = 2.0 * (cos(k) / ((k / l) * (k / (l * ((0.3333333333333333 / t) + ((1.0 / t) / (k * k)))))));
	} else {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-4.9d-17)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 4.6d-90) then
        tmp = 2.0d0 * (cos(k) / ((k / l) * (k / (l * ((0.3333333333333333d0 / t) + ((1.0d0 / t) / (k * k)))))))
    else
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.9e-17) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 4.6e-90) {
		tmp = 2.0 * (Math.cos(k) / ((k / l) * (k / (l * ((0.3333333333333333 / t) + ((1.0 / t) / (k * k)))))));
	} else {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -4.9e-17:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 4.6e-90:
		tmp = 2.0 * (math.cos(k) / ((k / l) * (k / (l * ((0.3333333333333333 / t) + ((1.0 / t) / (k * k)))))))
	else:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -4.9e-17)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 4.6e-90)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(k / l) * Float64(k / Float64(l * Float64(Float64(0.3333333333333333 / t) + Float64(Float64(1.0 / t) / Float64(k * k))))))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4.9e-17)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 4.6e-90)
		tmp = 2.0 * (cos(k) / ((k / l) * (k / (l * ((0.3333333333333333 / t) + ((1.0 / t) / (k * k)))))));
	else
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -4.9e-17], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-90], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k / N[(l * N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(N[(1.0 / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.90000000000000012e-17

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.6%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.7%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 85.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 54.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 54.7%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. associate-/l* 58.3%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

       3. unpow2 58.3%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

       4. associate-*l* 65.2%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   12. Simplified 65.2%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]

   if -4.90000000000000012e-17 < t < 4.5999999999999996e-90

   1. Initial program 45.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 45.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 45.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 45.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 45.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 45.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 77.9%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 76.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 76.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      2. fma-def 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      3. unpow2 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      4. associate-/l* 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{\ell}{\frac{t}{\ell}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}}\right)\right) \]

      6. associate-/r* 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}}\right)\right) \]

      7. unpow2 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}\right)\right) \]

      8. associate-/l* 74.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{2}}\right)\right) \]

      9. unpow2 74.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 74.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 69.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}\right)}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 69.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}}}} \]

       2. unpow2 69.3%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{\color{blue}{k \cdot k}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}}} \]

       3. *-commutative 69.3%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\color{blue}{{\ell}^{2} \cdot \left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}}} \]

       4. unpow2 69.3%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}} \]

       5. associate-*l* 75.0%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)\right)}}} \]

       6. associate-*r/ 75.0%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \frac{1}{{k}^{2} \cdot t}\right)\right)}} \]

       7. metadata-eval 75.0%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{0.3333333333333333}}{t} + \frac{1}{{k}^{2} \cdot t}\right)\right)}} \]

       8. *-commutative 75.0%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{1}{\color{blue}{t \cdot {k}^{2}}}\right)\right)}} \]

       9. associate-/r* 75.0%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \color{blue}{\frac{\frac{1}{t}}{{k}^{2}}}\right)\right)}} \]

       10. unpow2 75.0%

           \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{\color{blue}{k \cdot k}}\right)\right)}} \]

   12. Simplified 75.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{k \cdot k}\right)\right)}}} \]
   13. Step-by-step derivation

       1. times-frac 76.2%

          \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{k \cdot k}\right)}}} \]

   14. Applied egg-rr 76.2%

       \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{k \cdot k}\right)}}} \]

   if 4.5999999999999996e-90 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 66.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 66.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 66.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 66.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 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/ 77.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 77.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 88.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   10. Step-by-step derivation

       1. cbrt-div 96.5%

          \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Applied egg-rr 96.5%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   12. Taylor expanded in k around 0 55.5%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   13. Step-by-step derivation

       1. associate-/r* 56.6%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 56.6%

          \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

       3. unpow2 56.6%

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

       4. times-frac 69.1%

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   14. Simplified 69.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{k \cdot k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 14: 68.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\frac{\ell}{\frac{k \cdot k}{\ell}} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.4e-17)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 6.5e-87)
     (*
      2.0
      (*
       (/ (cos k) t)
       (/ (- (/ l (/ (* k k) l)) (* l (* l -0.3333333333333333))) (* k k))))
     (/ (* (/ l k) (/ l k)) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.4e-17) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 6.5e-87) {
		tmp = 2.0 * ((cos(k) / t) * (((l / ((k * k) / l)) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.4d-17)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 6.5d-87) then
        tmp = 2.0d0 * ((cos(k) / t) * (((l / ((k * k) / l)) - (l * (l * (-0.3333333333333333d0)))) / (k * k)))
    else
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.4e-17) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 6.5e-87) {
		tmp = 2.0 * ((Math.cos(k) / t) * (((l / ((k * k) / l)) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.4e-17:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 6.5e-87:
		tmp = 2.0 * ((math.cos(k) / t) * (((l / ((k * k) / l)) - (l * (l * -0.3333333333333333))) / (k * k)))
	else:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.4e-17)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 6.5e-87)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) * Float64(Float64(Float64(l / Float64(Float64(k * k) / l)) - Float64(l * Float64(l * -0.3333333333333333))) / Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.4e-17)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 6.5e-87)
		tmp = 2.0 * ((cos(k) / t) * (((l / ((k * k) / l)) - (l * (l * -0.3333333333333333))) / (k * k)));
	else
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.4e-17], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e-87], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3999999999999999e-17

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.6%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.7%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 85.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 54.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 54.7%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. associate-/l* 58.3%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

       3. unpow2 58.3%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

       4. associate-*l* 65.2%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   12. Simplified 65.2%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]

   if -1.3999999999999999e-17 < t < 6.5000000000000003e-87

   1. Initial program 45.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 45.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 45.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 45.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 45.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 45.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 77.9%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 76.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 76.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      2. fma-def 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      3. unpow2 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      4. associate-/l* 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{\ell}{\frac{t}{\ell}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}}\right)\right) \]

      6. associate-/r* 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}}\right)\right) \]

      7. unpow2 71.2%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}\right)\right) \]

      8. associate-/l* 74.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{2}}\right)\right) \]

      9. unpow2 74.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 74.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in t around -inf 69.5%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 69.5%

          \[\leadsto 2 \cdot \color{blue}{\left(-\frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]

       2. times-frac 71.0%

          \[\leadsto 2 \cdot \left(-\color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \frac{\cos k}{t}}\right) \]

       3. distribute-rgt-neg-in 71.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right)} \]

       4. mul-1-neg 71.0%

          \[\leadsto 2 \cdot \left(\frac{-0.3333333333333333 \cdot {\ell}^{2} + \color{blue}{\left(-\frac{{\ell}^{2}}{{k}^{2}}\right)}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       5. unsub-neg 71.0%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{-0.3333333333333333 \cdot {\ell}^{2} - \frac{{\ell}^{2}}{{k}^{2}}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       6. *-commutative 71.0%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -0.3333333333333333} - \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       7. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.3333333333333333 - \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       8. associate-*l* 71.0%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)} - \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       9. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       10. associate-/l* 76.5%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \color{blue}{\frac{\ell}{\frac{{k}^{2}}{\ell}}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       11. unpow2 76.5%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\ell}{\frac{\color{blue}{k \cdot k}}{\ell}}}{{k}^{2}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

       12. unpow2 76.5%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\ell}{\frac{k \cdot k}{\ell}}}{\color{blue}{k \cdot k}} \cdot \left(-\frac{\cos k}{t}\right)\right) \]

   12. Simplified 76.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - \frac{\ell}{\frac{k \cdot k}{\ell}}}{k \cdot k} \cdot \left(-\frac{\cos k}{t}\right)\right)} \]

   if 6.5000000000000003e-87 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 66.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 66.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 66.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 66.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 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/ 77.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 77.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 88.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   10. Step-by-step derivation

       1. cbrt-div 96.5%

          \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Applied egg-rr 96.5%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   12. Taylor expanded in k around 0 55.5%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   13. Step-by-step derivation

       1. associate-/r* 56.6%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 56.6%

          \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

       3. unpow2 56.6%

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

       4. times-frac 69.1%

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   14. Simplified 69.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\frac{\ell}{\frac{k \cdot k}{\ell}} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 15: 62.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 155000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 155000.0)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (* 2.0 (* 0.3333333333333333 (* (/ (cos k) (* k k)) (* l (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 155000.0) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} 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 <= 155000.0d0) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    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 <= 155000.0) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} 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 <= 155000.0:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	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 <= 155000.0)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	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 <= 155000.0)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	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, 155000.0], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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 < 155000

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 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* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 61.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.5%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 80.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 80.8%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 53.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 53.7%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. associate-/l* 57.9%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

       3. unpow2 57.9%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

       4. associate-*l* 64.0%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   12. Simplified 64.0%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]

   if 155000 < k

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 40.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 78.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 78.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 78.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      2. fma-def 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      3. unpow2 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      4. associate-/l* 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{\ell}{\frac{t}{\ell}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}}\right)\right) \]

      6. associate-/r* 69.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}}\right)\right) \]

      7. unpow2 69.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}\right)\right) \]

      8. associate-/l* 72.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{2}}\right)\right) \]

      9. unpow2 72.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 72.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 69.5%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. times-frac 69.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]

       2. unpow2 69.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       3. unpow2 69.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

       4. associate-*r/ 72.7%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right)\right) \]

   12. Simplified 72.7%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 155000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 16: 62.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 305000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{3}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 305000.0)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (* 2.0 (/ (cos k) (* (/ 3.0 l) (/ (* t (* k k)) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 305000.0) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 * (cos(k) / ((3.0 / 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 <= 305000.0d0) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else
        tmp = 2.0d0 * (cos(k) / ((3.0d0 / 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 <= 305000.0) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 * (Math.cos(k) / ((3.0 / l) * ((t * (k * k)) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 305000.0:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	else:
		tmp = 2.0 * (math.cos(k) / ((3.0 / l) * ((t * (k * k)) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 305000.0)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(3.0 / 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 <= 305000.0)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	else
		tmp = 2.0 * (cos(k) / ((3.0 / l) * ((t * (k * k)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 305000.0], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(3.0 / l), $MachinePrecision] * N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 305000

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 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* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 61.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.5%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 80.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 80.8%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 53.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 53.7%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. associate-/l* 57.9%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

       3. unpow2 57.9%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

       4. associate-*l* 64.0%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   12. Simplified 64.0%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]

   if 305000 < k

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 40.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 78.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 78.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 78.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      2. fma-def 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

      3. unpow2 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      4. associate-/l* 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{\ell}{\frac{t}{\ell}}}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 69.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}}\right)\right) \]

      6. associate-/r* 69.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}}\right)\right) \]

      7. unpow2 69.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}\right)\right) \]

      8. associate-/l* 72.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{2}}\right)\right) \]

      9. unpow2 72.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 72.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\ell}{\frac{t}{\ell}}, \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in l around 0 69.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}\right)}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 69.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}}}} \]

       2. unpow2 69.6%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{\color{blue}{k \cdot k}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}}} \]

       3. *-commutative 69.6%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\color{blue}{{\ell}^{2} \cdot \left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}}} \]

       4. unpow2 69.6%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)}} \]

       5. associate-*l* 72.7%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right)\right)}}} \]

       6. associate-*r/ 72.7%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \frac{1}{{k}^{2} \cdot t}\right)\right)}} \]

       7. metadata-eval 72.7%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{0.3333333333333333}}{t} + \frac{1}{{k}^{2} \cdot t}\right)\right)}} \]

       8. *-commutative 72.7%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{1}{\color{blue}{t \cdot {k}^{2}}}\right)\right)}} \]

       9. associate-/r* 72.7%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \color{blue}{\frac{\frac{1}{t}}{{k}^{2}}}\right)\right)}} \]

       10. unpow2 72.7%

           \[\leadsto 2 \cdot \frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{\color{blue}{k \cdot k}}\right)\right)}} \]

   12. Simplified 72.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{\cos k}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot \left(\frac{0.3333333333333333}{t} + \frac{\frac{1}{t}}{k \cdot k}\right)\right)}}} \]

   13. Taylor expanded in k around inf 69.5%

       \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{3 \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 69.5%

          \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{3 \cdot \left({k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]

       2. unpow2 69.5%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{3 \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

       3. times-frac 74.5%

          \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{3}{\ell} \cdot \frac{{k}^{2} \cdot t}{\ell}}} \]

       4. unpow2 74.5%

          \[\leadsto 2 \cdot \frac{\cos k}{\frac{3}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell}} \]

   15. Simplified 74.5%

       \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{3}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 305000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{3}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \]

Alternative 17: 65.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.12e-16)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 5.5e-91)
     (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
     (/ (* (/ l k) (/ l k)) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.12e-16) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 5.5e-91) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.12d-16)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 5.5d-91) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.12e-16) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 5.5e-91) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.12e-16:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 5.5e-91:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	else:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.12e-16)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 5.5e-91)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.12e-16)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 5.5e-91)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	else
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.12e-16], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-91], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.12e-16

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 62.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.6%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.6%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.6%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.6%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.7%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 85.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 85.9%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 54.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 54.7%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. associate-/l* 58.3%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

       3. unpow2 58.3%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

       4. associate-*l* 65.2%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   12. Simplified 65.2%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]

   if -1.12e-16 < t < 5.49999999999999965e-91

   1. Initial program 45.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 45.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 45.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 45.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 45.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 45.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 45.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 77.9%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 76.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 76.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 76.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 60.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 60.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   9. Simplified 60.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

   10. Taylor expanded in l around 0 60.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 60.7%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 60.7%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 67.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   12. Simplified 67.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 68.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   14. Applied egg-rr 68.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   if 5.49999999999999965e-91 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 66.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 66.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 66.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 66.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 66.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 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/ 77.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 77.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 88.9%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   10. Step-by-step derivation

       1. cbrt-div 96.5%

          \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   11. Applied egg-rr 96.5%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \left(\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\ell}\right)\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   12. Taylor expanded in k around 0 55.5%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   13. Step-by-step derivation

       1. associate-/r* 56.6%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 56.6%

          \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

       3. unpow2 56.6%

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

       4. times-frac 69.1%

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   14. Simplified 69.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 18: 56.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2000000:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2000000.0)
   (* (/ l (pow t 3.0)) (/ l (* k k)))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2000000.0) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2000000.0d0) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2000000.0) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2000000.0:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2000000.0)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2000000.0)
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2000000.0], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e6

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 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* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 62.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 61.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 61.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 53.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 53.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 53.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. times-frac 57.8%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   if 2e6 < k

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 40.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 78.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 78.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 78.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 60.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 60.8%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   9. Simplified 60.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

   10. Taylor expanded in l around 0 60.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 60.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 60.8%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 65.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   12. Simplified 65.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 65.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   14. Applied egg-rr 65.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 19: 57.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 29500000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{{t}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 29500000.0)
   (/ l (/ (* k k) (/ l (pow t 3.0))))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 29500000.0) {
		tmp = l / ((k * k) / (l / pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 29500000.0d0) then
        tmp = l / ((k * k) / (l / (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 29500000.0) {
		tmp = l / ((k * k) / (l / Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 29500000.0:
		tmp = l / ((k * k) / (l / math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 29500000.0)
		tmp = Float64(l / Float64(Float64(k * k) / Float64(l / (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 29500000.0)
		tmp = l / ((k * k) / (l / (t ^ 3.0)));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 29500000.0], N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.95e7

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 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* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 61.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.5%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Taylor expanded in k around 0 53.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   9. Step-by-step derivation

      1. *-commutative 53.7%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 53.8%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      3. unpow2 53.8%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}{{k}^{2}} \]

      4. associate-*r/ 57.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{{t}^{3}}}}{{k}^{2}} \]

      5. associate-/l* 57.9%

         \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2}}{\frac{\ell}{{t}^{3}}}}} \]

      6. unpow2 57.9%

         \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{{t}^{3}}}} \]

   10. Simplified 57.9%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{{t}^{3}}}}} \]

   if 2.95e7 < k

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 40.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 78.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 78.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 78.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 60.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 60.8%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   9. Simplified 60.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

   10. Taylor expanded in l around 0 60.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 60.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 60.8%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 65.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   12. Simplified 65.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 65.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   14. Applied egg-rr 65.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 29500000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{{t}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 20: 61.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7500000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7500000.0)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7500000.0) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7500000.0d0) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7500000.0) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7500000.0:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7500000.0)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7500000.0)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7500000.0], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.5e6

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 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* 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 61.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 61.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 61.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}\right) \cdot \sqrt[3]{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. pow-plus 75.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\left(2 + 1\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. metadata-eval 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{\color{blue}{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/r/ 75.5%

         \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}\right)}}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/r/ 75.5%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 75.5%

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \sqrt[3]{\frac{\ell}{\sin k} \cdot \ell}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. cbrt-prod 80.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Applied egg-rr 80.8%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \color{blue}{\left(\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}\right)}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 53.7%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 53.7%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. associate-/l* 57.9%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

       3. unpow2 57.9%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

       4. associate-*l* 64.0%

          \[\leadsto \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   12. Simplified 64.0%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]

   if 7.5e6 < k

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 40.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 40.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 40.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 78.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 78.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 78.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 78.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 60.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 60.8%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   9. Simplified 60.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

   10. Taylor expanded in l around 0 60.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 60.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 60.8%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 65.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   12. Simplified 65.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   13. Step-by-step derivation

       1. associate-*l/ 65.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   14. Applied egg-rr 65.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7500000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 21: 55.5% 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 56.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 51.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 51.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.0%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 56.3%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 55.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 55.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 55.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 61.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

   2. unpow2 60.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

   3. unpow2 60.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

   4. *-commutative 60.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

6. Simplified 60.7%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 53.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 53.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 53.5%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

9. Simplified 53.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

10. Taylor expanded in l around 0 53.5%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 53.5%

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

    2. *-commutative 53.5%

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

    3. times-frac 57.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

12. Simplified 57.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

13. Final simplification 57.8%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Alternative 22: 55.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (/ l (pow k 4.0))) t)))

C

double code(double t, double l, double k) {
	return 2.0 * ((l * (l / pow(k, 4.0))) / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
}

Python

def code(t, l, k):
	return 2.0 * ((l * (l / math.pow(k, 4.0))) / t)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 56.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 51.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 51.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 56.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 56.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 56.0%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 56.3%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 55.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 55.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 55.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 61.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. times-frac 60.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

   2. unpow2 60.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

   3. unpow2 60.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

   4. *-commutative 60.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

6. Simplified 60.7%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 53.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 53.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 53.5%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

9. Simplified 53.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

10. Taylor expanded in l around 0 53.5%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 53.5%

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

    2. *-commutative 53.5%

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

    3. times-frac 57.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

12. Simplified 57.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
13. Step-by-step derivation

    1. associate-*l/ 58.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

14. Applied egg-rr 58.0%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

15. Final simplification 58.0%

    \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \]

Reproduce

herbie shell --seed 2023274 
(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))))