Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 85.4%

Time: 22.7s

Alternatives: 11

Speedup: 2.0×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% 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: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{2}}{{t}^{1.5}}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{t_2}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sin k \cdot t_2} \cdot \frac{t_1}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* l (sqrt 2.0)) (pow t 1.5))) (t_2 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -2.8e-76)
     (/
      (/ 2.0 (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (tan k)))
      t_2)
     (if (<= t 2e-89)
       (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) (/ l (tan k)))
       (* (/ t_1 (* (sin k) 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 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -2.8e-76) {
		tmp = (2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * tan(k))) / t_2;
	} else if (t <= 2e-89) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * (l / tan(k));
	} else {
		tmp = (t_1 / (sin(k) * 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 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -2.8e-76) {
		tmp = (2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * Math.tan(k))) / t_2;
	} else if (t <= 2e-89) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * (l / Math.tan(k));
	} else {
		tmp = (t_1 / (Math.sin(k) * t_2)) * (t_1 / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l * sqrt(2.0)) / (t ^ 1.5))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -2.8e-76)
		tmp = Float64(Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * tan(k))) / t_2);
	elseif (t <= 2e-89)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64(t_1 / Float64(sin(k) * t_2)) * Float64(t_1 / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e-76], N[(N[(2.0 / 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]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t, 2e-89], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.8000000000000001e-76

   1. Initial program 65.1%

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

   3. Simplified 72.5%

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

      1. add-cube-cbrt 72.6%

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

      2. pow3 72.6%

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

      3. cbrt-div 72.4%

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

      4. rem-cbrt-cube 78.5%

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

   5. Applied egg-rr 78.5%

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

      1. add-cube-cbrt 78.4%

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

      2. pow2 78.4%

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

      3. cbrt-div 78.4%

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

      4. unpow3 78.4%

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

      5. add-cbrt-cube 78.4%

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

      6. cbrt-div 78.4%

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

      7. unpow3 78.4%

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

      8. add-cbrt-cube 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. add-cube-cbrt 88.5%

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

      2. pow3 88.5%

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

      3. cbrt-prod 88.5%

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

      4. unpow2 88.5%

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

      5. add-cbrt-cube 95.6%

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

      6. associate-/l/ 95.7%

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

      7. pow2 95.7%

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

   9. Applied egg-rr 95.7%

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

   if -2.8000000000000001e-76 < t < 2.00000000000000008e-89

   1. Initial program 41.6%

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

   3. Simplified 41.3%

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

      1. associate-*r* 47.2%

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

      2. associate-*r* 47.2%

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

      3. times-frac 48.4%

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

      4. div-inv 48.4%

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

      5. pow-flip 48.4%

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

      6. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in t around 0 85.3%

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

   if 2.00000000000000008e-89 < t

   1. Initial program 65.0%

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

   3. Simplified 55.9%

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

      1. add-sqr-sqrt 55.9%

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

      2. associate-*r* 55.9%

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

      3. times-frac 64.9%

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

      4. *-commutative 64.9%

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

      5. sqrt-prod 65.0%

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

      6. sqrt-prod 30.8%

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

      7. add-sqr-sqrt 53.2%

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

      8. sqrt-div 53.2%

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

      9. sqrt-pow1 53.2%

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

      10. metadata-eval 53.2%

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

   5. Applied egg-rr 88.1%

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

      1. associate-*r/ 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-*r/ 88.1%

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

   7. Simplified 88.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \sqrt{2}}{{t}^{1.5}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\frac{\ell \cdot \sqrt{2}}{{t}^{1.5}}}{\tan k}\\ \end{array} \]

Alternative 2: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(t_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t_1 \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + t_2} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))) (t_2 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_2 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+287)
     (* t_1 (* (/ (* 2.0 (pow t -3.0)) (+ 2.0 t_2)) (/ l (sin k))))
     (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) t_1))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double t_2 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+287) {
		tmp = t_1 * (((2.0 * pow(t, -3.0)) / (2.0 + t_2)) * (l / sin(k)));
	} else {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l / tan(k)
    t_2 = (k / t) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (t_2 + 1.0d0)) * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 5d+287) then
        tmp = t_1 * (((2.0d0 * (t ** (-3.0d0))) / (2.0d0 + t_2)) * (l / sin(k)))
    else
        tmp = (2.0d0 * (l / ((k ** 2.0d0) * (t * sin(k))))) * t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double t_2 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+287) {
		tmp = t_1 * (((2.0 * Math.pow(t, -3.0)) / (2.0 + t_2)) * (l / Math.sin(k)));
	} else {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.tan(k)
	t_2 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (t_2 + 1.0)) * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 5e+287:
		tmp = t_1 * (((2.0 * math.pow(t, -3.0)) / (2.0 + t_2)) * (l / math.sin(k)))
	else:
		tmp = (2.0 * (l / (math.pow(k, 2.0) * (t * math.sin(k))))) * t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	t_2 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+287)
		tmp = Float64(t_1 * Float64(Float64(Float64(2.0 * (t ^ -3.0)) / Float64(2.0 + t_2)) * Float64(l / sin(k))));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	t_2 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 5e+287)
		tmp = t_1 * (((2.0 * (t ^ -3.0)) / (2.0 + t_2)) * (l / sin(k)));
	else
		tmp = (2.0 * (l / ((k ^ 2.0) * (t * sin(k))))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+287], N[(t$95$1 * N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $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))) < 5e287

   1. Initial program 79.6%

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

   3. Simplified 68.9%

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

      1. associate-*r* 70.8%

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

      2. associate-*r* 70.8%

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

      3. times-frac 86.2%

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

      4. div-inv 86.2%

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

      5. pow-flip 86.6%

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

      6. metadata-eval 86.6%

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

   5. Applied egg-rr 86.6%

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

      1. times-frac 89.2%

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

   7. Simplified 89.2%

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

   if 5e287 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 23.0%

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

   3. Simplified 23.8%

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

      1. associate-*r* 33.7%

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

      2. associate-*r* 33.7%

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

      3. times-frac 33.9%

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

      4. div-inv 33.9%

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

      5. pow-flip 33.9%

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

      6. metadata-eval 33.9%

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

   5. Applied egg-rr 33.9%

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

   6. Taylor expanded in t around 0 64.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \]

Alternative 3: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-73} \lor \neg \left(t \leq 4.4 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6e-73) (not (<= t 4.4e-85)))
   (/
    (/ 2.0 (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (tan k)))
    (+ 2.0 (pow (/ k t) 2.0)))
   (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) (/ l (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6e-73) || !(t <= 4.4e-85)) {
		tmp = (2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * tan(k))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * (l / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6e-73) || !(t <= 4.4e-85)) {
		tmp = (2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * Math.tan(k))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * (l / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6e-73) || !(t <= 4.4e-85))
		tmp = Float64(Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * tan(k))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * Float64(l / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6e-73], N[Not[LessEqual[t, 4.4e-85]], $MachinePrecision]], N[(N[(2.0 / 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]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6e-73 or 4.4e-85 < t

   1. Initial program 65.0%

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

   3. Simplified 70.7%

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

      1. add-cube-cbrt 70.6%

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

      2. pow3 70.6%

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

      3. cbrt-div 70.6%

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

      4. rem-cbrt-cube 75.1%

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

   5. Applied egg-rr 75.1%

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

      1. add-cube-cbrt 75.0%

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

      2. pow2 75.0%

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

      3. cbrt-div 75.0%

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

      4. unpow3 75.0%

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

      5. add-cbrt-cube 75.0%

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

      6. cbrt-div 74.9%

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

      7. unpow3 74.9%

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

      8. add-cbrt-cube 83.7%

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

   7. Applied egg-rr 83.7%

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

      1. add-cube-cbrt 83.6%

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

      2. pow3 83.6%

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

      3. cbrt-prod 83.6%

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

      4. unpow2 83.6%

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

      5. add-cbrt-cube 90.2%

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

      6. associate-/l/ 90.2%

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

      7. pow2 90.2%

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

   9. Applied egg-rr 90.2%

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

   if -6e-73 < t < 4.4e-85

   1. Initial program 41.6%

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

   3. Simplified 41.3%

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

      1. associate-*r* 47.2%

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

      2. associate-*r* 47.2%

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

      3. times-frac 48.4%

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

      4. div-inv 48.4%

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

      5. pow-flip 48.4%

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

      6. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in t around 0 85.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-73} \lor \neg \left(t \leq 4.4 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \]

Alternative 4: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{t}{\sqrt[3]{\ell}}\\ t_3 := \frac{\ell}{\tan k}\\ t_4 := \frac{2 \cdot {t}^{-3}}{t_1} \cdot \frac{\ell}{\sin k}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{t_1}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{\ell \cdot t_4}{\tan k}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-86}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot t_3\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+105}:\\ \;\;\;\;t_3 \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(t_2 \cdot \frac{{t_2}^{2}}{\ell}\right)\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (/ t (cbrt l)))
        (t_3 (/ l (tan k)))
        (t_4 (* (/ (* 2.0 (pow t -3.0)) t_1) (/ l (sin k)))))
   (if (<= t -6.8e+117)
     (/ (/ 2.0 (* (tan k) (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0)))) t_1)
     (if (<= t -2.3e-76)
       (/ (* l t_4) (tan k))
       (if (<= t 8.6e-86)
         (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) t_3)
         (if (<= t 1.2e+105)
           (* t_3 t_4)
           (/
            (/ 2.0 (* (tan k) (* (sin k) (* t_2 (/ (pow t_2 2.0) l)))))
            t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = t / cbrt(l);
	double t_3 = l / tan(k);
	double t_4 = ((2.0 * pow(t, -3.0)) / t_1) * (l / sin(k));
	double tmp;
	if (t <= -6.8e+117) {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)))) / t_1;
	} else if (t <= -2.3e-76) {
		tmp = (l * t_4) / tan(k);
	} else if (t <= 8.6e-86) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * t_3;
	} else if (t <= 1.2e+105) {
		tmp = t_3 * t_4;
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * (t_2 * (pow(t_2, 2.0) / l))))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = t / Math.cbrt(l);
	double t_3 = l / Math.tan(k);
	double t_4 = ((2.0 * Math.pow(t, -3.0)) / t_1) * (l / Math.sin(k));
	double tmp;
	if (t <= -6.8e+117) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)))) / t_1;
	} else if (t <= -2.3e-76) {
		tmp = (l * t_4) / Math.tan(k);
	} else if (t <= 8.6e-86) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * t_3;
	} else if (t <= 1.2e+105) {
		tmp = t_3 * t_4;
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * (t_2 * (Math.pow(t_2, 2.0) / l))))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(t / cbrt(l))
	t_3 = Float64(l / tan(k))
	t_4 = Float64(Float64(Float64(2.0 * (t ^ -3.0)) / t_1) * Float64(l / sin(k)))
	tmp = 0.0
	if (t <= -6.8e+117)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)))) / t_1);
	elseif (t <= -2.3e-76)
		tmp = Float64(Float64(l * t_4) / tan(k));
	elseif (t <= 8.6e-86)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * t_3);
	elseif (t <= 1.2e+105)
		tmp = Float64(t_3 * t_4);
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(t_2 * Float64((t_2 ^ 2.0) / l))))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+117], 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] / t$95$1), $MachinePrecision], If[LessEqual[t, -2.3e-76], N[(N[(l * t$95$4), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.6e-86], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t, 1.2e+105], N[(t$95$3 * t$95$4), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$2 * N[(N[Power[t$95$2, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.8000000000000002e117

   1. Initial program 61.6%

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

   3. Simplified 66.5%

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

      1. add-cube-cbrt 66.5%

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

      2. pow3 66.5%

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

      3. cbrt-div 66.5%

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

      4. rem-cbrt-cube 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. add-cube-cbrt 77.3%

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

      2. pow2 77.3%

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

      3. cbrt-div 77.3%

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

      4. unpow3 77.3%

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

      5. add-cbrt-cube 77.3%

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

      6. cbrt-div 77.3%

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

      7. unpow3 77.3%

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

      8. add-cbrt-cube 95.8%

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

   7. Applied egg-rr 95.8%

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

      1. expm1-log1p-u 59.3%

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

      2. expm1-udef 44.0%

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

      3. pow-plus 44.0%

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

      4. associate-/l/ 44.0%

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

      5. pow2 44.0%

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

      6. metadata-eval 44.0%

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

   9. Applied egg-rr 44.0%

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

       1. expm1-def 59.3%

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

       2. expm1-log1p 95.8%

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

       3. *-commutative 95.8%

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

   11. Simplified 95.8%

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

   if -6.8000000000000002e117 < t < -2.30000000000000006e-76

   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 66.5%

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

      1. associate-*r* 74.6%

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

      2. associate-*r* 74.6%

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

      3. times-frac 88.6%

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

      4. div-inv 88.6%

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

      5. pow-flip 88.7%

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

      6. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. associate-*r/ 88.7%

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

      2. times-frac 91.5%

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

   7. Applied egg-rr 91.5%

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

   if -2.30000000000000006e-76 < t < 8.60000000000000026e-86

   1. Initial program 41.6%

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

   3. Simplified 41.3%

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

      1. associate-*r* 47.2%

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

      2. associate-*r* 47.2%

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

      3. times-frac 48.4%

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

      4. div-inv 48.4%

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

      5. pow-flip 48.4%

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

      6. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in t around 0 85.3%

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

   if 8.60000000000000026e-86 < t < 1.19999999999999987e105

   1. Initial program 71.6%

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

   3. Simplified 69.2%

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

      1. associate-*r* 71.6%

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

      2. associate-*r* 71.6%

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

      3. times-frac 81.9%

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

      4. div-inv 81.9%

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

      5. pow-flip 83.0%

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

      6. metadata-eval 83.0%

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

   5. Applied egg-rr 83.0%

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

      1. times-frac 87.7%

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

   7. Simplified 87.7%

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

   if 1.19999999999999987e105 < t

   1. Initial program 57.7%

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

   3. Simplified 62.1%

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

      1. add-cube-cbrt 62.1%

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

      2. pow3 62.1%

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

      3. cbrt-div 62.1%

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

      4. rem-cbrt-cube 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. unpow3 68.5%

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

      2. unpow2 68.5%

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

      3. associate-*r/ 84.3%

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

      4. associate-/l/ 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l/ 63.7%

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

      3. *-commutative 63.7%

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

      4. times-frac 84.3%

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

   9. Simplified 84.3%

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-86}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{\ell}\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 5: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{t_1}\\ t_3 := \frac{\ell}{\tan k}\\ t_4 := \frac{2 \cdot {t}^{-3}}{t_1} \cdot \frac{\ell}{\sin k}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\ell \cdot t_4}{\tan k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-89}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot t_3\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;t_3 \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2
         (/
          (/ 2.0 (* (tan k) (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
          t_1))
        (t_3 (/ l (tan k)))
        (t_4 (* (/ (* 2.0 (pow t -3.0)) t_1) (/ l (sin k)))))
   (if (<= t -6.8e+117)
     t_2
     (if (<= t -3.2e-76)
       (/ (* l t_4) (tan k))
       (if (<= t 8e-89)
         (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) t_3)
         (if (<= t 1.75e+105) (* t_3 t_4) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = (2.0 / (tan(k) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)))) / t_1;
	double t_3 = l / tan(k);
	double t_4 = ((2.0 * pow(t, -3.0)) / t_1) * (l / sin(k));
	double tmp;
	if (t <= -6.8e+117) {
		tmp = t_2;
	} else if (t <= -3.2e-76) {
		tmp = (l * t_4) / tan(k);
	} else if (t <= 8e-89) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * t_3;
	} else if (t <= 1.75e+105) {
		tmp = t_3 * t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)))) / t_1;
	double t_3 = l / Math.tan(k);
	double t_4 = ((2.0 * Math.pow(t, -3.0)) / t_1) * (l / Math.sin(k));
	double tmp;
	if (t <= -6.8e+117) {
		tmp = t_2;
	} else if (t <= -3.2e-76) {
		tmp = (l * t_4) / Math.tan(k);
	} else if (t <= 8e-89) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * t_3;
	} else if (t <= 1.75e+105) {
		tmp = t_3 * t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)))) / t_1)
	t_3 = Float64(l / tan(k))
	t_4 = Float64(Float64(Float64(2.0 * (t ^ -3.0)) / t_1) * Float64(l / sin(k)))
	tmp = 0.0
	if (t <= -6.8e+117)
		tmp = t_2;
	elseif (t <= -3.2e-76)
		tmp = Float64(Float64(l * t_4) / tan(k));
	elseif (t <= 8e-89)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * t_3);
	elseif (t <= 1.75e+105)
		tmp = Float64(t_3 * t_4);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+117], t$95$2, If[LessEqual[t, -3.2e-76], N[(N[(l * t$95$4), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-89], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t, 1.75e+105], N[(t$95$3 * t$95$4), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.8000000000000002e117 or 1.74999999999999996e105 < t

   1. Initial program 59.6%

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

   3. Simplified 64.3%

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

      1. add-cube-cbrt 64.3%

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

      2. pow3 64.3%

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

      3. cbrt-div 64.3%

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

      4. rem-cbrt-cube 72.9%

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

   5. Applied egg-rr 72.9%

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

      1. add-cube-cbrt 72.9%

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

      2. pow2 72.9%

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

      3. cbrt-div 72.9%

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

      4. unpow3 72.9%

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

      5. add-cbrt-cube 72.9%

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

      6. cbrt-div 72.8%

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

      7. unpow3 72.8%

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

      8. add-cbrt-cube 89.9%

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

   7. Applied egg-rr 89.9%

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

      1. expm1-log1p-u 48.7%

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

      2. expm1-udef 37.8%

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

      3. pow-plus 37.8%

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

      4. associate-/l/ 37.8%

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

      5. pow2 37.8%

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

      6. metadata-eval 37.8%

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

   9. Applied egg-rr 37.8%

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

       1. expm1-def 48.7%

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

       2. expm1-log1p 90.0%

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

       3. *-commutative 90.0%

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

   11. Simplified 90.0%

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

   if -6.8000000000000002e117 < t < -3.1999999999999998e-76

   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 66.5%

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

      1. associate-*r* 74.6%

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

      2. associate-*r* 74.6%

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

      3. times-frac 88.6%

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

      4. div-inv 88.6%

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

      5. pow-flip 88.7%

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

      6. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. associate-*r/ 88.7%

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

      2. times-frac 91.5%

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

   7. Applied egg-rr 91.5%

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

   if -3.1999999999999998e-76 < t < 8.00000000000000031e-89

   1. Initial program 41.6%

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

   3. Simplified 41.3%

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

      1. associate-*r* 47.2%

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

      2. associate-*r* 47.2%

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

      3. times-frac 48.4%

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

      4. div-inv 48.4%

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

      5. pow-flip 48.4%

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

      6. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in t around 0 85.3%

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

   if 8.00000000000000031e-89 < t < 1.74999999999999996e105

   1. Initial program 71.6%

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

   3. Simplified 69.2%

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

      1. associate-*r* 71.6%

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

      2. associate-*r* 71.6%

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

      3. times-frac 81.9%

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

      4. div-inv 81.9%

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

      5. pow-flip 83.0%

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

      6. metadata-eval 83.0%

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

   5. Applied egg-rr 83.0%

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

      1. times-frac 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-89}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 6: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := \frac{\ell}{\tan k}\\ t_4 := \frac{2 \cdot {t}^{-3}}{t_2} \cdot \frac{\ell}{\sin k}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{t_2}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{\ell \cdot t_4}{\tan k}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot t_3\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;t_3 \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2 (+ 2.0 t_1))
        (t_3 (/ l (tan k)))
        (t_4 (* (/ (* 2.0 (pow t -3.0)) t_2) (/ l (sin k)))))
   (if (<= t -6.8e+117)
     (/ (/ 2.0 (* (tan k) (* (sin k) (/ (pow (/ t (cbrt l)) 3.0) l)))) t_2)
     (if (<= t -3.3e-74)
       (/ (* l t_4) (tan k))
       (if (<= t 4e-86)
         (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) t_3)
         (if (<= t 2.15e+105)
           (* t_3 t_4)
           (/
            2.0
            (*
             (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))
             (+ 1.0 (+ t_1 1.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = 2.0 + t_1;
	double t_3 = l / tan(k);
	double t_4 = ((2.0 * pow(t, -3.0)) / t_2) * (l / sin(k));
	double tmp;
	if (t <= -6.8e+117) {
		tmp = (2.0 / (tan(k) * (sin(k) * (pow((t / cbrt(l)), 3.0) / l)))) / t_2;
	} else if (t <= -3.3e-74) {
		tmp = (l * t_4) / tan(k);
	} else if (t <= 4e-86) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * t_3;
	} else if (t <= 2.15e+105) {
		tmp = t_3 * t_4;
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))) * (1.0 + (t_1 + 1.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = 2.0 + t_1;
	double t_3 = l / Math.tan(k);
	double t_4 = ((2.0 * Math.pow(t, -3.0)) / t_2) * (l / Math.sin(k));
	double tmp;
	if (t <= -6.8e+117) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * (Math.pow((t / Math.cbrt(l)), 3.0) / l)))) / t_2;
	} else if (t <= -3.3e-74) {
		tmp = (l * t_4) / Math.tan(k);
	} else if (t <= 4e-86) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * t_3;
	} else if (t <= 2.15e+105) {
		tmp = t_3 * t_4;
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) * (1.0 + (t_1 + 1.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(2.0 + t_1)
	t_3 = Float64(l / tan(k))
	t_4 = Float64(Float64(Float64(2.0 * (t ^ -3.0)) / t_2) * Float64(l / sin(k)))
	tmp = 0.0
	if (t <= -6.8e+117)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)))) / t_2);
	elseif (t <= -3.3e-74)
		tmp = Float64(Float64(l * t_4) / tan(k));
	elseif (t <= 4e-86)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * t_3);
	elseif (t <= 2.15e+105)
		tmp = Float64(t_3 * t_4);
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(t_1 + 1.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+117], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t, -3.3e-74], N[(N[(l * t$95$4), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-86], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t, 2.15e+105], N[(t$95$3 * t$95$4), $MachinePrecision], N[(2.0 / N[(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] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.8000000000000002e117

   1. Initial program 61.6%

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

   3. Simplified 66.5%

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

      1. add-cube-cbrt 66.5%

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

      2. pow3 66.5%

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

      3. cbrt-div 66.5%

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

      4. rem-cbrt-cube 77.3%

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

   5. Applied egg-rr 77.3%

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

   if -6.8000000000000002e117 < t < -3.29999999999999996e-74

   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 66.5%

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

      1. associate-*r* 74.6%

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

      2. associate-*r* 74.6%

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

      3. times-frac 88.6%

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

      4. div-inv 88.6%

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

      5. pow-flip 88.7%

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

      6. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. associate-*r/ 88.7%

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

      2. times-frac 91.5%

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

   7. Applied egg-rr 91.5%

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

   if -3.29999999999999996e-74 < t < 4.00000000000000034e-86

   1. Initial program 41.6%

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

   3. Simplified 41.3%

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

      1. associate-*r* 47.2%

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

      2. associate-*r* 47.2%

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

      3. times-frac 48.4%

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

      4. div-inv 48.4%

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

      5. pow-flip 48.4%

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

      6. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in t around 0 85.3%

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

   if 4.00000000000000034e-86 < t < 2.1500000000000001e105

   1. Initial program 71.6%

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

   3. Simplified 69.2%

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

      1. associate-*r* 71.6%

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

      2. associate-*r* 71.6%

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

      3. times-frac 81.9%

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

      4. div-inv 81.9%

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

      5. pow-flip 83.0%

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

      6. metadata-eval 83.0%

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

   5. Applied egg-rr 83.0%

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

      1. times-frac 87.7%

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

   7. Simplified 87.7%

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

   if 2.1500000000000001e105 < t

   1. Initial program 57.7%

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

      1. add-sqr-sqrt 57.7%

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

      2. pow2 57.7%

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

      3. sqrt-div 57.7%

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

      4. sqrt-pow1 66.7%

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

      5. metadata-eval 66.7%

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

      6. sqrt-prod 37.7%

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

      7. add-sqr-sqrt 81.6%

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

   3. Applied egg-rr 81.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{2 \cdot {t}^{-3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \end{array} \]

Alternative 7: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-86}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))))
   (if (<= t -1.75e+55)
     (* t_1 (/ l (* (sin k) (pow t 3.0))))
     (if (<= t 1.9e-86)
       (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) 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 / tan(k);
	double tmp;
	if (t <= -1.75e+55) {
		tmp = t_1 * (l / (sin(k) * pow(t, 3.0)));
	} else if (t <= 1.9e-86) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * 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.tan(k);
	double tmp;
	if (t <= -1.75e+55) {
		tmp = t_1 * (l / (Math.sin(k) * Math.pow(t, 3.0)));
	} else if (t <= 1.9e-86) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * 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.tan(k)
	tmp = 0
	if t <= -1.75e+55:
		tmp = t_1 * (l / (math.sin(k) * math.pow(t, 3.0)))
	elif t <= 1.9e-86:
		tmp = (2.0 * (l / (math.pow(k, 2.0) * (t * math.sin(k))))) * 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(l / tan(k))
	tmp = 0.0
	if (t <= -1.75e+55)
		tmp = Float64(t_1 * Float64(l / Float64(sin(k) * (t ^ 3.0))));
	elseif (t <= 1.9e-86)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * 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 / tan(k);
	tmp = 0.0;
	if (t <= -1.75e+55)
		tmp = t_1 * (l / (sin(k) * (t ^ 3.0)));
	elseif (t <= 1.9e-86)
		tmp = (2.0 * (l / ((k ^ 2.0) * (t * sin(k))))) * 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[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.75e+55], N[(t$95$1 * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-86], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -1.75000000000000005e55

   1. Initial program 62.8%

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

   3. Simplified 51.3%

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

      1. associate-*r* 57.7%

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

      2. associate-*r* 57.7%

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

      3. times-frac 72.4%

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

      4. div-inv 72.4%

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

      5. pow-flip 72.5%

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

      6. metadata-eval 72.5%

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

   5. Applied egg-rr 72.5%

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

   6. Taylor expanded in t around inf 74.0%

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

   if -1.75000000000000005e55 < t < 1.9e-86

   1. Initial program 47.1%

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

   3. Simplified 46.8%

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

      1. associate-*r* 52.6%

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

      2. associate-*r* 52.6%

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

      3. times-frac 55.5%

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

      4. div-inv 55.5%

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

      5. pow-flip 55.5%

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

      6. metadata-eval 55.5%

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

   5. Applied egg-rr 55.5%

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

   6. Taylor expanded in t around 0 83.2%

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

   if 1.9e-86 < t

   1. Initial program 65.0%

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

      1. add-sqr-sqrt 50.8%

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

      2. pow2 50.8%

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

   3. Applied egg-rr 53.7%

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

   4. Taylor expanded in k around 0 76.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-86}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 8: 67.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;k \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))))
   (if (<= k 2.5e-28)
     (* t_1 (/ (/ l k) (pow t 3.0)))
     (* t_1 (* 2.0 (/ l (* (sin k) (* t (pow k 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double tmp;
	if (k <= 2.5e-28) {
		tmp = t_1 * ((l / k) / pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (sin(k) * (t * 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) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k)
    if (k <= 2.5d-28) then
        tmp = t_1 * ((l / k) / (t ** 3.0d0))
    else
        tmp = t_1 * (2.0d0 * (l / (sin(k) * (t * (k ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double tmp;
	if (k <= 2.5e-28) {
		tmp = t_1 * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (Math.sin(k) * (t * Math.pow(k, 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.tan(k)
	tmp = 0
	if k <= 2.5e-28:
		tmp = t_1 * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = t_1 * (2.0 * (l / (math.sin(k) * (t * math.pow(k, 2.0)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	tmp = 0.0
	if (k <= 2.5e-28)
		tmp = Float64(t_1 * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(l / Float64(sin(k) * Float64(t * (k ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	tmp = 0.0;
	if (k <= 2.5e-28)
		tmp = t_1 * ((l / k) / (t ^ 3.0));
	else
		tmp = t_1 * (2.0 * (l / (sin(k) * (t * (k ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.5e-28], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(l / N[(N[Sin[k], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000001e-28

   1. Initial program 60.0%

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

   3. Simplified 52.0%

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

      1. associate-*r* 57.0%

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

      2. associate-*r* 57.0%

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

      3. times-frac 68.8%

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

      4. div-inv 68.8%

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

      5. pow-flip 69.1%

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

      6. metadata-eval 69.1%

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

   5. Applied egg-rr 69.1%

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

   6. Taylor expanded in k around 0 66.9%

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

      1. associate-/r* 68.2%

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

   8. Simplified 68.2%

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

   if 2.5000000000000001e-28 < k

   1. Initial program 48.7%

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

   3. Simplified 48.5%

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

      1. associate-*r* 53.8%

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

      2. associate-*r* 53.8%

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

      3. times-frac 55.4%

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

      4. div-inv 55.4%

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

      5. pow-flip 55.4%

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

      6. metadata-eval 55.4%

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

   5. Applied egg-rr 55.4%

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

   6. Taylor expanded in t around 0 70.7%

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

      1. associate-*r* 70.6%

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

   8. Simplified 70.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \end{array} \]

Alternative 9: 67.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;k \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))))
   (if (<= k 1.85e-28)
     (* t_1 (/ (/ l k) (pow t 3.0)))
     (* (* 2.0 (/ l (* (pow k 2.0) (* t (sin k))))) t_1))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double tmp;
	if (k <= 1.85e-28) {
		tmp = t_1 * ((l / k) / pow(t, 3.0));
	} else {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t * sin(k))))) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k)
    if (k <= 1.85d-28) then
        tmp = t_1 * ((l / k) / (t ** 3.0d0))
    else
        tmp = (2.0d0 * (l / ((k ** 2.0d0) * (t * sin(k))))) * t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double tmp;
	if (k <= 1.85e-28) {
		tmp = t_1 * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t * Math.sin(k))))) * t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.tan(k)
	tmp = 0
	if k <= 1.85e-28:
		tmp = t_1 * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = (2.0 * (l / (math.pow(k, 2.0) * (t * math.sin(k))))) * t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	tmp = 0.0
	if (k <= 1.85e-28)
		tmp = Float64(t_1 * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t * sin(k))))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	tmp = 0.0;
	if (k <= 1.85e-28)
		tmp = t_1 * ((l / k) / (t ^ 3.0));
	else
		tmp = (2.0 * (l / ((k ^ 2.0) * (t * sin(k))))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.85e-28], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8500000000000001e-28

   1. Initial program 60.0%

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

   3. Simplified 52.0%

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

      1. associate-*r* 57.0%

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

      2. associate-*r* 57.0%

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

      3. times-frac 68.8%

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

      4. div-inv 68.8%

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

      5. pow-flip 69.1%

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

      6. metadata-eval 69.1%

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

   5. Applied egg-rr 69.1%

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

   6. Taylor expanded in k around 0 66.9%

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

      1. associate-/r* 68.2%

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

   8. Simplified 68.2%

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

   if 1.8500000000000001e-28 < k

   1. Initial program 48.7%

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

   3. Simplified 48.5%

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

      1. associate-*r* 53.8%

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

      2. associate-*r* 53.8%

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

      3. times-frac 55.4%

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

      4. div-inv 55.4%

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

      5. pow-flip 55.4%

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

      6. metadata-eval 55.4%

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

   5. Applied egg-rr 55.4%

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

   6. Taylor expanded in t around 0 70.7%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \]

Alternative 10: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot {t}^{3}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (tan k)) (/ l (* k (pow t 3.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Simplified 51.1%

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

   1. associate-*r* 56.2%

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

   2. associate-*r* 56.2%

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

   3. times-frac 65.6%

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

   4. div-inv 65.6%

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

   5. pow-flip 65.8%

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

   6. metadata-eval 65.8%

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

5. Applied egg-rr 65.8%

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

6. Taylor expanded in k around 0 61.2%

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

7. Final simplification 61.2%

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

Alternative 11: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (tan k)) (/ (/ l k) (pow t 3.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Simplified 51.1%

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

   1. associate-*r* 56.2%

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

   2. associate-*r* 56.2%

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

   3. times-frac 65.6%

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

   4. div-inv 65.6%

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

   5. pow-flip 65.8%

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

   6. metadata-eval 65.8%

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

5. Applied egg-rr 65.8%

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

6. Taylor expanded in k around 0 61.2%

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

   1. associate-/r* 62.6%

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

8. Simplified 62.6%

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

9. Final simplification 62.6%

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

Reproduce

herbie shell --seed 2023308 
(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))))