Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 86.7%

Time: 20.3s

Alternatives: 14

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\sqrt{2}}{{t}^{1.5}}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + t_2\right)\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sin k \cdot \left(2 + t_2\right)} \cdot \frac{t_1}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (/ (sqrt 2.0) (pow t 1.5)))) (t_2 (pow (/ k t) 2.0)))
   (if (<= t -2.5e-94)
     (/
      2.0
      (*
       (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (tan k))
       (+ 1.0 (+ 1.0 t_2))))
     (if (<= t 2.2e-85)
       (* (/ (/ (* 2.0 l) (pow k 2.0)) t) (/ (/ l (sin k)) (tan k)))
       (* (/ t_1 (* (sin k) (+ 2.0 t_2))) (/ t_1 (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = l * (sqrt(2.0) / pow(t, 1.5));
	double t_2 = pow((k / t), 2.0);
	double tmp;
	if (t <= -2.5e-94) {
		tmp = 2.0 / ((pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * tan(k)) * (1.0 + (1.0 + t_2)));
	} else if (t <= 2.2e-85) {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (t_1 / (sin(k) * (2.0 + t_2))) * (t_1 / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (Math.sqrt(2.0) / Math.pow(t, 1.5));
	double t_2 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -2.5e-94) {
		tmp = 2.0 / ((Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * Math.tan(k)) * (1.0 + (1.0 + t_2)));
	} else if (t <= 2.2e-85) {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (t_1 / (Math.sin(k) * (2.0 + t_2))) * (t_1 / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(sqrt(2.0) / (t ^ 1.5)))
	t_2 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -2.5e-94)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * tan(k)) * Float64(1.0 + Float64(1.0 + t_2))));
	elseif (t <= 2.2e-85)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(t_1 / Float64(sin(k) * Float64(2.0 + t_2))) * Float64(t_1 / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -2.5e-94], N[(2.0 / N[(N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e-85], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.4999999999999998e-94

   1. Initial program 69.9%

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

      1. associate-/r* 74.3%

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

      2. add-cube-cbrt 74.3%

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

      3. *-un-lft-identity 74.3%

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

      4. times-frac 74.2%

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

      5. pow2 74.2%

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

      6. cbrt-div 74.3%

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

      7. rem-cbrt-cube 74.4%

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

      8. cbrt-div 74.3%

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

      9. rem-cbrt-cube 84.9%

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

   3. Applied egg-rr 84.9%

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

      1. add-cube-cbrt 84.8%

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

      2. pow3 84.8%

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

      3. frac-times 78.2%

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

      4. unpow2 78.2%

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

      5. *-un-lft-identity 78.2%

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

      6. cbrt-div 78.3%

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

      7. add-cbrt-cube 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. pow1/3 39.1%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{{\ell}^{0.3333333333333333}}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. add-sqr-sqrt 39.1%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow-prod-down 39.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{{\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied egg-rr 39.2%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{{\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. unpow1/3 39.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. unpow1/3 40.0%

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

   9. Simplified 40.0%

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

       1. add-cube-cbrt 40.0%

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

       2. pow3 40.0%

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

   11. Applied egg-rr 88.7%

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

   if -2.4999999999999998e-94 < t < 2.2e-85

   1. Initial program 36.4%

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

   3. Simplified 37.3%

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

      1. associate-*r* 43.6%

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

      2. *-un-lft-identity 43.6%

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

      3. times-frac 43.6%

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

   5. Applied egg-rr 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. associate-*r/ 43.6%

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

      3. times-frac 43.6%

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

      4. *-commutative 43.6%

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

      5. associate-/r* 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-/r* 87.8%

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

      4. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   if 2.2e-85 < t

   1. Initial program 70.7%

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

   3. Simplified 62.6%

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

      1. add-sqr-sqrt 62.6%

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

      2. associate-*r* 62.6%

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

      3. times-frac 70.6%

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

      4. *-commutative 70.6%

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

      5. sqrt-prod 70.6%

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

      6. sqrt-prod 40.2%

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

      7. add-sqr-sqrt 60.4%

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

      8. sqrt-div 60.4%

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

      9. sqrt-pow1 60.4%

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

      10. metadata-eval 60.4%

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

   5. Applied egg-rr 92.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\sqrt{2}}{{t}^{1.5}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell \cdot \frac{\sqrt{2}}{{t}^{1.5}}}{\tan k}\\ \end{array} \]

Alternative 2: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+198}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* t_1 (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+198)
     (/ 2.0 (* t_1 (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
     (* (/ (/ (* 2.0 l) (pow k 2.0)) t) (/ (/ l (sin k)) (tan k))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / (t_1 * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+198) {
		tmp = 2.0 / (t_1 * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if ((2.0d0 / (t_1 * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 5d+198) then
        tmp = 2.0d0 / (t_1 * (tan(k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l)))))
    else
        tmp = (((2.0d0 * l) / (k ** 2.0d0)) / t) * ((l / sin(k)) / tan(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / (t_1 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+198) {
		tmp = 2.0 / (t_1 * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if (2.0 / (t_1 * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 5e+198:
		tmp = 2.0 / (t_1 * (math.tan(k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l)))))
	else:
		tmp = (((2.0 * l) / math.pow(k, 2.0)) / t) * ((l / math.sin(k)) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+198)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / (t_1 * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 5e+198)
		tmp = 2.0 / (t_1 * (tan(k) * (sin(k) * (((t ^ 2.0) / l) * (t / l)))));
	else
		tmp = (((2.0 * l) / (k ^ 2.0)) / t) * ((l / sin(k)) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+198], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.00000000000000049e198

   1. Initial program 85.8%

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

      1. unpow3 85.8%

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

      2. times-frac 91.9%

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

      3. pow2 91.9%

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

   3. Applied egg-rr 91.9%

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

   if 5.00000000000000049e198 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 22.2%

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

   3. Simplified 23.0%

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

      1. associate-*r* 32.3%

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

      2. *-un-lft-identity 32.3%

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

      3. times-frac 33.1%

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

   5. Applied egg-rr 33.1%

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

      1. /-rgt-identity 33.1%

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

      2. associate-*r/ 32.3%

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

      3. times-frac 33.1%

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

      4. *-commutative 33.1%

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

      5. associate-/r* 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in t around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-/r* 74.5%

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

      4. *-commutative 74.5%

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

   10. Simplified 74.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+198}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 3: 79.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $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))) < +inf.0

   1. Initial program 85.6%

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

   3. Simplified 79.0%

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

      1. associate-*r* 80.4%

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

      2. associate-*r* 80.5%

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

      3. times-frac 89.4%

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

   5. Applied egg-rr 89.4%

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

   if +inf.0 < (/.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 0.0%

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

   3. Simplified 0.0%

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

      1. associate-*r* 11.5%

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

      2. *-un-lft-identity 11.5%

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

      3. times-frac 11.5%

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

   5. Applied egg-rr 11.5%

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

      1. /-rgt-identity 11.5%

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

      2. associate-*r/ 11.5%

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

      3. times-frac 11.5%

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

      4. *-commutative 11.5%

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

      5. associate-/r* 11.5%

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

   7. Simplified 11.5%

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

   8. Taylor expanded in t around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. *-commutative 67.1%

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

      3. associate-/r* 67.7%

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

      4. *-commutative 67.7%

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

   10. Simplified 67.7%

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

4. Final simplification 82.3%

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

Alternative 4: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<= t -2e-94)
     (/
      2.0
      (*
       (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (tan k))
       t_1))
     (if (<= t 1.4e-85)
       (* (/ (/ (* 2.0 l) (pow k 2.0)) t) (/ (/ l (sin k)) (tan k)))
       (/
        2.0
        (*
         t_1
         (* (tan k) (* (sin k) (pow (/ (/ t (cbrt l)) (cbrt l)) 3.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if (t <= -2e-94) {
		tmp = 2.0 / ((pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * tan(k)) * t_1);
	} else if (t <= 1.4e-85) {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * ((l / sin(k)) / tan(k));
	} else {
		tmp = 2.0 / (t_1 * (tan(k) * (sin(k) * pow(((t / cbrt(l)) / cbrt(l)), 3.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if (t <= -2e-94) {
		tmp = 2.0 / ((Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * Math.tan(k)) * t_1);
	} else if (t <= 1.4e-85) {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = 2.0 / (t_1 * (Math.tan(k) * (Math.sin(k) * Math.pow(((t / Math.cbrt(l)) / Math.cbrt(l)), 3.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (t <= -2e-94)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * tan(k)) * t_1));
	elseif (t <= 1.4e-85)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(sin(k) * (Float64(Float64(t / cbrt(l)) / cbrt(l)) ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e-94], N[(2.0 / N[(N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-85], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999999e-94

   1. Initial program 69.9%

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

      1. associate-/r* 74.3%

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

      2. add-cube-cbrt 74.3%

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

      3. *-un-lft-identity 74.3%

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

      4. times-frac 74.2%

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

      5. pow2 74.2%

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

      6. cbrt-div 74.3%

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

      7. rem-cbrt-cube 74.4%

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

      8. cbrt-div 74.3%

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

      9. rem-cbrt-cube 84.9%

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

   3. Applied egg-rr 84.9%

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

      1. add-cube-cbrt 84.8%

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

      2. pow3 84.8%

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

      3. frac-times 78.2%

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

      4. unpow2 78.2%

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

      5. *-un-lft-identity 78.2%

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

      6. cbrt-div 78.3%

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

      7. add-cbrt-cube 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. pow1/3 39.1%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{{\ell}^{0.3333333333333333}}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. add-sqr-sqrt 39.1%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow-prod-down 39.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{{\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied egg-rr 39.2%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{{\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. unpow1/3 39.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}\right)}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. unpow1/3 40.0%

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

   9. Simplified 40.0%

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

       1. add-cube-cbrt 40.0%

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

       2. pow3 40.0%

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

   11. Applied egg-rr 88.7%

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

   if -1.9999999999999999e-94 < t < 1.40000000000000008e-85

   1. Initial program 36.4%

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

   3. Simplified 37.3%

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

      1. associate-*r* 43.6%

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

      2. *-un-lft-identity 43.6%

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

      3. times-frac 43.6%

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

   5. Applied egg-rr 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. associate-*r/ 43.6%

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

      3. times-frac 43.6%

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

      4. *-commutative 43.6%

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

      5. associate-/r* 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-/r* 87.8%

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

      4. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   if 1.40000000000000008e-85 < t

   1. Initial program 70.7%

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

      1. associate-/r* 75.8%

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

      2. add-cube-cbrt 75.6%

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

      3. *-un-lft-identity 75.6%

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

      4. times-frac 75.6%

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

      5. pow2 75.6%

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

      6. cbrt-div 75.6%

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

      7. rem-cbrt-cube 75.7%

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

      8. cbrt-div 75.6%

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

      9. rem-cbrt-cube 88.7%

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

   3. Applied egg-rr 88.7%

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

      1. add-cube-cbrt 88.7%

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

      2. pow3 88.7%

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

      3. frac-times 85.6%

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

      4. unpow2 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. cbrt-div 85.4%

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

      7. add-cbrt-cube 91.8%

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

   5. Applied egg-rr 91.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)\right)}\\ \end{array} \]

Alternative 5: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-94} \lor \neg \left(t \leq 8.6 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.15e-94) (not (<= t 8.6e-86)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (* (sin k) (pow (/ (/ t (cbrt l)) (cbrt l)) 3.0)))))
   (* (/ (/ (* 2.0 l) (pow k 2.0)) t) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.15e-94) || !(t <= 8.6e-86)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * pow(((t / cbrt(l)) / cbrt(l)), 3.0))));
	} else {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.15e-94) || !(t <= 8.6e-86)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * Math.pow(((t / Math.cbrt(l)) / Math.cbrt(l)), 3.0))));
	} else {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.15e-94) || !(t <= 8.6e-86))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * (Float64(Float64(t / cbrt(l)) / cbrt(l)) ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2.15e-94], N[Not[LessEqual[t, 8.6e-86]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1499999999999999e-94 or 8.60000000000000026e-86 < t

   1. Initial program 70.3%

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

      1. associate-/r* 75.1%

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

      2. add-cube-cbrt 75.0%

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

      3. *-un-lft-identity 75.0%

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

      4. times-frac 75.0%

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

      5. pow2 75.0%

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

      6. cbrt-div 75.0%

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

      7. rem-cbrt-cube 75.1%

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

      8. cbrt-div 75.0%

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

      9. rem-cbrt-cube 87.0%

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

   3. Applied egg-rr 87.0%

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

      1. add-cube-cbrt 87.0%

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

      2. pow3 87.0%

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

      3. frac-times 82.3%

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

      4. unpow2 82.3%

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

      5. *-un-lft-identity 82.3%

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

      6. cbrt-div 82.2%

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

      7. add-cbrt-cube 89.3%

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

   5. Applied egg-rr 89.3%

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

   if -2.1499999999999999e-94 < t < 8.60000000000000026e-86

   1. Initial program 36.4%

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

   3. Simplified 37.3%

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

      1. associate-*r* 43.6%

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

      2. *-un-lft-identity 43.6%

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

      3. times-frac 43.6%

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

   5. Applied egg-rr 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. associate-*r/ 43.6%

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

      3. times-frac 43.6%

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

      4. *-commutative 43.6%

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

      5. associate-/r* 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-/r* 87.8%

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

      4. *-commutative 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-94} \lor \neg \left(t \leq 8.6 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 6: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-94} \lor \neg \left(t \leq 2.25 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.5e-94) (not (<= t 2.25e-85)))
   (/
    (/ 2.0 (* (tan k) (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (* (/ (/ (* 2.0 l) (pow k 2.0)) t) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.5e-94) || !(t <= 2.25e-85)) {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.5e-94) || !(t <= 2.25e-85)) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.5e-94) || !(t <= 2.25e-85))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2.5e-94], N[Not[LessEqual[t, 2.25e-85]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4999999999999998e-94 or 2.25000000000000002e-85 < t

   1. Initial program 70.3%

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

   3. Simplified 75.1%

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

      1. add-cube-cbrt 75.0%

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

      2. pow3 75.0%

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

      3. cbrt-div 75.0%

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

      4. rem-cbrt-cube 82.3%

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

   5. Applied egg-rr 82.3%

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

      1. add-cube-cbrt 82.3%

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

      2. pow3 82.3%

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

      3. cbrt-div 82.2%

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

      4. unpow3 82.2%

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

      5. add-cbrt-cube 89.3%

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

      6. associate-/l/ 89.2%

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

      7. pow2 89.2%

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

   7. Applied egg-rr 89.2%

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

   if -2.4999999999999998e-94 < t < 2.25000000000000002e-85

   1. Initial program 36.4%

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

   3. Simplified 37.3%

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

      1. associate-*r* 43.6%

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

      2. *-un-lft-identity 43.6%

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

      3. times-frac 43.6%

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

   5. Applied egg-rr 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. associate-*r/ 43.6%

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

      3. times-frac 43.6%

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

      4. *-commutative 43.6%

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

      5. associate-/r* 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-/r* 87.8%

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

      4. *-commutative 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-94} \lor \neg \left(t \leq 2.25 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 7: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt[3]{\ell}}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({t_1}^{2} \cdot \frac{t_1}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cbrt l))))
   (if (<= t -2.5e-94)
     (/
      2.0
      (*
       (* (tan k) (* (sin k) (* (pow t_1 2.0) (/ t_1 l))))
       (+ 1.0 (+ 1.0 (* (/ k t) (/ k t))))))
     (if (<= t 2.2e-85)
       (* (/ (/ (* 2.0 l) (pow k 2.0)) t) (/ (/ l (sin k)) (tan k)))
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
         (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = t / cbrt(l);
	double tmp;
	if (t <= -2.5e-94) {
		tmp = 2.0 / ((tan(k) * (sin(k) * (pow(t_1, 2.0) * (t_1 / l)))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	} else if (t <= 2.2e-85) {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * ((l / sin(k)) / tan(k));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.cbrt(l);
	double tmp;
	if (t <= -2.5e-94) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_1, 2.0) * (t_1 / l)))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	} else if (t <= 2.2e-85) {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / cbrt(l))
	tmp = 0.0
	if (t <= -2.5e-94)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_1 ^ 2.0) * Float64(t_1 / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t))))));
	elseif (t <= 2.2e-85)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e-94], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e-85], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.4999999999999998e-94

   1. Initial program 69.9%

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

      1. associate-/r* 74.3%

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

      2. add-cube-cbrt 74.3%

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

      3. *-un-lft-identity 74.3%

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

      4. times-frac 74.2%

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

      5. pow2 74.2%

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

      6. cbrt-div 74.3%

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

      7. rem-cbrt-cube 74.4%

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

      8. cbrt-div 74.3%

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

      9. rem-cbrt-cube 84.9%

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

   3. Applied egg-rr 84.9%

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

      1. unpow2 84.9%

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

   5. Applied egg-rr 84.9%

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

   if -2.4999999999999998e-94 < t < 2.2e-85

   1. Initial program 36.4%

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

   3. Simplified 37.3%

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

      1. associate-*r* 43.6%

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

      2. *-un-lft-identity 43.6%

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

      3. times-frac 43.6%

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

   5. Applied egg-rr 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. associate-*r/ 43.6%

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

      3. times-frac 43.6%

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

      4. *-commutative 43.6%

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

      5. associate-/r* 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-/r* 87.8%

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

      4. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   if 2.2e-85 < t

   1. Initial program 70.7%

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

      1. add-sqr-sqrt 70.6%

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

      2. pow2 70.6%

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

      3. sqrt-div 70.7%

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

      4. sqrt-pow1 79.7%

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

      5. metadata-eval 79.7%

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

      6. sqrt-prod 50.6%

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

      7. add-sqr-sqrt 89.0%

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

   3. Applied egg-rr 89.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 8: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l (sin k)) (tan k))))
   (if (<= t -2.5e-94)
     (* t_1 (/ (* l (/ 2.0 (pow t 3.0))) (+ 2.0 (pow (/ k t) 2.0))))
     (if (<= t 5.8e-82)
       (* (/ (/ (* 2.0 l) (pow k 2.0)) t) t_1)
       (/
        2.0
        (pow
         (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t))))
         2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / sin(k)) / tan(k);
	double tmp;
	if (t <= -2.5e-94) {
		tmp = t_1 * ((l * (2.0 / pow(t, 3.0))) / (2.0 + pow((k / t), 2.0)));
	} else if (t <= 5.8e-82) {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * t_1;
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / Math.sin(k)) / Math.tan(k);
	double tmp;
	if (t <= -2.5e-94) {
		tmp = t_1 * ((l * (2.0 / Math.pow(t, 3.0))) / (2.0 + Math.pow((k / t), 2.0)));
	} else if (t <= 5.8e-82) {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * t_1;
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / math.sin(k)) / math.tan(k)
	tmp = 0
	if t <= -2.5e-94:
		tmp = t_1 * ((l * (2.0 / math.pow(t, 3.0))) / (2.0 + math.pow((k / t), 2.0)))
	elif t <= 5.8e-82:
		tmp = (((2.0 * l) / math.pow(k, 2.0)) / t) * t_1
	else:
		tmp = 2.0 / math.pow(((k * (math.pow(t, 1.5) / l)) * math.hypot(1.0, math.hypot(1.0, (k / t)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if (t <= -2.5e-94)
		tmp = Float64(t_1 * Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(2.0 + (Float64(k / t) ^ 2.0))));
	elseif (t <= 5.8e-82)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * t_1);
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / sin(k)) / tan(k);
	tmp = 0.0;
	if (t <= -2.5e-94)
		tmp = t_1 * ((l * (2.0 / (t ^ 3.0))) / (2.0 + ((k / t) ^ 2.0)));
	elseif (t <= 5.8e-82)
		tmp = (((2.0 * l) / (k ^ 2.0)) / t) * t_1;
	else
		tmp = 2.0 / (((k * ((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e-94], N[(t$95$1 * N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e-82], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.4999999999999998e-94

   1. Initial program 69.9%

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

   3. Simplified 62.9%

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

      1. associate-*r* 66.5%

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

      2. *-un-lft-identity 66.5%

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

      3. times-frac 67.7%

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

   5. Applied egg-rr 67.7%

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

      1. /-rgt-identity 67.7%

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

      2. associate-*r/ 66.5%

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

      3. times-frac 67.7%

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

      4. *-commutative 67.7%

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

      5. associate-/r* 73.9%

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

   7. Simplified 73.9%

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

   if -2.4999999999999998e-94 < t < 5.79999999999999954e-82

   1. Initial program 37.7%

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

   3. Simplified 38.5%

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

      1. associate-*r* 44.7%

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

      2. *-un-lft-identity 44.7%

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

      3. times-frac 44.7%

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

   5. Applied egg-rr 44.7%

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

      1. /-rgt-identity 44.7%

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

      2. associate-*r/ 44.7%

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

      3. times-frac 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-/r* 44.8%

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

   7. Simplified 44.8%

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

   8. Taylor expanded in t around 0 87.6%

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

      1. associate-*r/ 87.6%

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

      2. *-commutative 87.6%

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

      3. associate-/r* 88.1%

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

      4. *-commutative 88.1%

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

   10. Simplified 88.1%

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

   if 5.79999999999999954e-82 < t

   1. Initial program 70.0%

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

      1. add-sqr-sqrt 57.0%

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

      2. pow2 57.0%

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

   3. Applied egg-rr 65.8%

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

   4. Taylor expanded in k around 0 80.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 9: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -3.75 \cdot 10^{+18}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l (sin k)) (tan k))))
   (if (<= t -3.75e+18)
     (* t_1 (/ l (pow t 3.0)))
     (if (<= t 7.8e-82)
       (* (/ (/ (* 2.0 l) (pow k 2.0)) t) t_1)
       (/
        2.0
        (pow
         (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t))))
         2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / sin(k)) / tan(k);
	double tmp;
	if (t <= -3.75e+18) {
		tmp = t_1 * (l / pow(t, 3.0));
	} else if (t <= 7.8e-82) {
		tmp = (((2.0 * l) / pow(k, 2.0)) / t) * t_1;
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / Math.sin(k)) / Math.tan(k);
	double tmp;
	if (t <= -3.75e+18) {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	} else if (t <= 7.8e-82) {
		tmp = (((2.0 * l) / Math.pow(k, 2.0)) / t) * t_1;
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / math.sin(k)) / math.tan(k)
	tmp = 0
	if t <= -3.75e+18:
		tmp = t_1 * (l / math.pow(t, 3.0))
	elif t <= 7.8e-82:
		tmp = (((2.0 * l) / math.pow(k, 2.0)) / t) * t_1
	else:
		tmp = 2.0 / math.pow(((k * (math.pow(t, 1.5) / l)) * math.hypot(1.0, math.hypot(1.0, (k / t)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if (t <= -3.75e+18)
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	elseif (t <= 7.8e-82)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (k ^ 2.0)) / t) * t_1);
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / sin(k)) / tan(k);
	tmp = 0.0;
	if (t <= -3.75e+18)
		tmp = t_1 * (l / (t ^ 3.0));
	elseif (t <= 7.8e-82)
		tmp = (((2.0 * l) / (k ^ 2.0)) / t) * t_1;
	else
		tmp = 2.0 / (((k * ((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.75e+18], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e-82], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.75e18

   1. Initial program 70.8%

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

   3. Simplified 62.0%

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

      1. associate-*r* 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. times-frac 67.1%

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

   5. Applied egg-rr 67.1%

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

      1. /-rgt-identity 67.1%

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

      2. associate-*r/ 65.7%

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

      3. times-frac 67.1%

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

      4. *-commutative 67.1%

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

      5. associate-/r* 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in t around inf 70.8%

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

   if -3.75e18 < t < 7.79999999999999947e-82

   1. Initial program 41.5%

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

   3. Simplified 42.2%

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

      1. associate-*r* 48.0%

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

      2. *-un-lft-identity 48.0%

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

      3. times-frac 48.0%

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

   5. Applied egg-rr 48.0%

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

      1. /-rgt-identity 48.0%

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

      2. associate-*r/ 48.0%

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

      3. times-frac 48.0%

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

      4. *-commutative 48.0%

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

      5. associate-/r* 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in t around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. *-commutative 84.2%

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

      3. associate-/r* 84.5%

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

      4. *-commutative 84.5%

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

   10. Simplified 84.5%

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

   if 7.79999999999999947e-82 < t

   1. Initial program 70.0%

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

      1. add-sqr-sqrt 57.0%

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

      2. pow2 57.0%

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

   3. Applied egg-rr 65.8%

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

   4. Taylor expanded in k around 0 80.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.75 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{k}^{2}}}{t} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 10: 68.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;k \leq 9.6 \cdot 10^{-231} \lor \neg \left(k \leq 2.5 \cdot 10^{-49}\right):\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l (sin k)) (tan k))))
   (if (or (<= k 9.6e-231) (not (<= k 2.5e-49)))
     (* t_1 (* 2.0 (/ l (* t (pow k 2.0)))))
     (* t_1 (/ l (pow t 3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / sin(k)) / tan(k);
	double tmp;
	if ((k <= 9.6e-231) || !(k <= 2.5e-49)) {
		tmp = t_1 * (2.0 * (l / (t * pow(k, 2.0))));
	} else {
		tmp = t_1 * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / sin(k)) / tan(k)
    if ((k <= 9.6d-231) .or. (.not. (k <= 2.5d-49))) then
        tmp = t_1 * (2.0d0 * (l / (t * (k ** 2.0d0))))
    else
        tmp = t_1 * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / Math.sin(k)) / Math.tan(k);
	double tmp;
	if ((k <= 9.6e-231) || !(k <= 2.5e-49)) {
		tmp = t_1 * (2.0 * (l / (t * Math.pow(k, 2.0))));
	} else {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / math.sin(k)) / math.tan(k)
	tmp = 0
	if (k <= 9.6e-231) or not (k <= 2.5e-49):
		tmp = t_1 * (2.0 * (l / (t * math.pow(k, 2.0))))
	else:
		tmp = t_1 * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if ((k <= 9.6e-231) || !(k <= 2.5e-49))
		tmp = Float64(t_1 * Float64(2.0 * Float64(l / Float64(t * (k ^ 2.0)))));
	else
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / sin(k)) / tan(k);
	tmp = 0.0;
	if ((k <= 9.6e-231) || ~((k <= 2.5e-49)))
		tmp = t_1 * (2.0 * (l / (t * (k ^ 2.0))));
	else
		tmp = t_1 * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, 9.6e-231], N[Not[LessEqual[k, 2.5e-49]], $MachinePrecision]], N[(t$95$1 * N[(2.0 * N[(l / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.59999999999999967e-231 or 2.4999999999999999e-49 < k

   1. Initial program 55.4%

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

   3. Simplified 52.1%

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

      1. associate-*r* 56.5%

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

      2. *-un-lft-identity 56.5%

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

      3. times-frac 57.4%

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

   5. Applied egg-rr 57.4%

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

      1. /-rgt-identity 57.4%

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

      2. associate-*r/ 56.5%

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

      3. times-frac 57.4%

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

      4. *-commutative 57.4%

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

      5. associate-/r* 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in t around 0 72.6%

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

   if 9.59999999999999967e-231 < k < 2.4999999999999999e-49

   1. Initial program 69.4%

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

   3. Simplified 58.7%

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

      1. associate-*r* 65.2%

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

      2. *-un-lft-identity 65.2%

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

      3. times-frac 64.9%

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

   5. Applied egg-rr 64.9%

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

      1. /-rgt-identity 64.9%

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

      2. associate-*r/ 65.2%

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

      3. times-frac 64.9%

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

      4. *-commutative 64.9%

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

      5. associate-/r* 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in t around inf 79.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-231} \lor \neg \left(k \leq 2.5 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 11: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.95e+122)
   (* (/ (/ l (sin k)) (tan k)) (/ l (pow t 3.0)))
   (* 2.0 (/ (* (pow k -4.0) (pow l 2.0)) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.95e+122) {
		tmp = ((l / sin(k)) / tan(k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((pow(k, -4.0) * pow(l, 2.0)) / 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 <= 2.95d+122) then
        tmp = ((l / sin(k)) / tan(k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (((k ** (-4.0d0)) * (l ** 2.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.95e+122) {
		tmp = ((l / Math.sin(k)) / Math.tan(k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((Math.pow(k, -4.0) * Math.pow(l, 2.0)) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.95e+122:
		tmp = ((l / math.sin(k)) / math.tan(k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((math.pow(k, -4.0) * math.pow(l, 2.0)) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.95e+122)
		tmp = Float64(Float64(Float64(l / sin(k)) / tan(k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((k ^ -4.0) * (l ^ 2.0)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.95e+122)
		tmp = ((l / sin(k)) / tan(k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * (((k ^ -4.0) * (l ^ 2.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.95e+122], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[k, -4.0], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.95000000000000016e122

   1. Initial program 59.9%

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

   3. Simplified 54.8%

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

      1. associate-*r* 59.9%

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

      2. *-un-lft-identity 59.9%

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

      3. times-frac 60.2%

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

   5. Applied egg-rr 60.2%

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

      1. /-rgt-identity 60.2%

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

      2. associate-*r/ 59.9%

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

      3. times-frac 60.2%

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

      4. *-commutative 60.2%

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

      5. associate-/r* 64.6%

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

   7. Simplified 64.6%

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

   8. Taylor expanded in t around inf 63.1%

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

   if 2.95000000000000016e122 < k

   1. Initial program 43.8%

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

      1. add-sqr-sqrt 28.4%

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

      2. pow2 28.4%

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

   3. Applied egg-rr 28.1%

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

   4. Taylor expanded in k around inf 51.1%

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

      1. *-commutative 51.1%

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

      2. associate-/l* 51.0%

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

      3. associate-*r/ 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in k around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-/r* 61.9%

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

   9. Simplified 61.9%

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

       1. expm1-log1p-u 61.9%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

       2. expm1-udef 61.9%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

       3. div-inv 61.9%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

       4. pow-flip 61.9%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

       5. metadata-eval 61.9%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

   11. Applied egg-rr 61.9%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 61.9%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-log1p 61.9%

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

       3. *-commutative 61.9%

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

       4. associate-*r/ 62.5%

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

   13. Simplified 62.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{{k}^{-4} \cdot {\ell}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t}\\ \end{array} \]

Alternative 12: 53.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-171} \lor \neg \left(k \leq 1.65 \cdot 10^{-50}\right):\\ \;\;\;\;2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot {t}^{-3}\right)}{2 \cdot {k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k 1.7e-171) (not (<= k 1.65e-50)))
   (* 2.0 (/ (* (pow k -4.0) (pow l 2.0)) t))
   (/ (* (* l l) (* 2.0 (pow t -3.0))) (* 2.0 (pow k 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= 1.7e-171) || !(k <= 1.65e-50)) {
		tmp = 2.0 * ((pow(k, -4.0) * pow(l, 2.0)) / t);
	} else {
		tmp = ((l * l) * (2.0 * pow(t, -3.0))) / (2.0 * pow(k, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= 1.7d-171) .or. (.not. (k <= 1.65d-50))) then
        tmp = 2.0d0 * (((k ** (-4.0d0)) * (l ** 2.0d0)) / t)
    else
        tmp = ((l * l) * (2.0d0 * (t ** (-3.0d0)))) / (2.0d0 * (k ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 1.7e-171) || !(k <= 1.65e-50)) {
		tmp = 2.0 * ((Math.pow(k, -4.0) * Math.pow(l, 2.0)) / t);
	} else {
		tmp = ((l * l) * (2.0 * Math.pow(t, -3.0))) / (2.0 * Math.pow(k, 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= 1.7e-171) or not (k <= 1.65e-50):
		tmp = 2.0 * ((math.pow(k, -4.0) * math.pow(l, 2.0)) / t)
	else:
		tmp = ((l * l) * (2.0 * math.pow(t, -3.0))) / (2.0 * math.pow(k, 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= 1.7e-171) || !(k <= 1.65e-50))
		tmp = Float64(2.0 * Float64(Float64((k ^ -4.0) * (l ^ 2.0)) / t));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 * (t ^ -3.0))) / Float64(2.0 * (k ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= 1.7e-171) || ~((k <= 1.65e-50)))
		tmp = 2.0 * (((k ^ -4.0) * (l ^ 2.0)) / t);
	else
		tmp = ((l * l) * (2.0 * (t ^ -3.0))) / (2.0 * (k ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, 1.7e-171], N[Not[LessEqual[k, 1.65e-50]], $MachinePrecision]], N[(2.0 * N[(N[(N[Power[k, -4.0], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.69999999999999993e-171 or 1.6499999999999999e-50 < k

   1. Initial program 56.5%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

   3. Applied egg-rr 27.2%

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

   4. Taylor expanded in k around inf 37.2%

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

      1. *-commutative 37.2%

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

      2. associate-/l* 37.2%

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

      3. associate-*r/ 36.8%

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

   6. Simplified 36.8%

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

   7. Taylor expanded in k around 0 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 56.2%

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

   9. Simplified 56.2%

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

       1. expm1-log1p-u 43.7%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

       2. expm1-udef 44.9%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

       3. div-inv 44.5%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

       4. pow-flip 44.5%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

       5. metadata-eval 44.5%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

   11. Applied egg-rr 44.5%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 43.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-log1p 55.8%

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

       3. *-commutative 55.8%

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

       4. associate-*r/ 56.0%

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

   13. Simplified 56.0%

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

   if 1.69999999999999993e-171 < k < 1.6499999999999999e-50

   1. Initial program 65.4%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in k around 0 65.3%

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

      1. div-inv 65.3%

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

      2. pow-flip 68.9%

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

      3. metadata-eval 68.9%

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

   6. Applied egg-rr 68.9%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-171} \lor \neg \left(k \leq 1.65 \cdot 10^{-50}\right):\\ \;\;\;\;2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot {t}^{-3}\right)}{2 \cdot {k}^{2}}\\ \end{array} \]

Alternative 13: 51.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* (pow k -4.0) (pow l 2.0)) t)))

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(k, -4.0) * pow(l, 2.0)) / 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 * (((k ** (-4.0d0)) * (l ** 2.0d0)) / t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((Math.pow(k, -4.0) * Math.pow(l, 2.0)) / t);
}

Python

def code(t, l, k):
	return 2.0 * ((math.pow(k, -4.0) * math.pow(l, 2.0)) / t)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64((k ^ -4.0) * (l ^ 2.0)) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((k ^ -4.0) * (l ^ 2.0)) / t);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[k, -4.0], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

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

   1. add-sqr-sqrt 29.2%

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

   2. pow2 29.2%

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

3. Applied egg-rr 30.0%

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

4. Taylor expanded in k around inf 37.2%

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

   1. *-commutative 37.2%

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

   2. associate-/l* 37.2%

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

   3. associate-*r/ 36.8%

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

6. Simplified 36.8%

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

7. Taylor expanded in k around 0 54.8%

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

   1. *-commutative 54.8%

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

   2. associate-/r* 56.1%

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

9. Simplified 56.1%

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

    1. expm1-log1p-u 43.3%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

    2. expm1-udef 44.8%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

    3. div-inv 44.4%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

    4. pow-flip 44.4%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

    5. metadata-eval 44.4%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

11. Applied egg-rr 44.4%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 42.9%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

    2. expm1-log1p 55.7%

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

    3. *-commutative 55.7%

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

    4. associate-*r/ 55.5%

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

13. Simplified 55.5%

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

14. Final simplification 55.5%

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

Alternative 14: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (/ (pow l 2.0) t) (pow k 4.0))))

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) / t) / 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 ** 2.0d0) / t) / (k ** 4.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((Math.pow(l, 2.0) / t) / Math.pow(k, 4.0));
}

Python

def code(t, l, k):
	return 2.0 * ((math.pow(l, 2.0) / t) / math.pow(k, 4.0))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (k ^ 4.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) / t) / (k ^ 4.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

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

   1. add-sqr-sqrt 29.2%

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

   2. pow2 29.2%

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

3. Applied egg-rr 30.0%

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

4. Taylor expanded in k around inf 37.2%

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

   1. *-commutative 37.2%

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

   2. associate-/l* 37.2%

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

   3. associate-*r/ 36.8%

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

6. Simplified 36.8%

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

7. Taylor expanded in k around 0 54.8%

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

   1. *-commutative 54.8%

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

   2. associate-/r* 56.1%

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

9. Simplified 56.1%

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

10. Final simplification 56.1%

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

Reproduce

herbie shell --seed 2023309 
(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))))