Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 90.7%

Time: 22.4s

Alternatives: 15

Speedup: 1.3×

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-51} \lor \neg \left(t \leq 2.1 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.8e-51) (not (<= t 2.1e-98)))
   (/
    2.0
    (pow
     (*
      (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k)))
      (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
     3.0))
   (* (* (/ l (pow k 2.0)) (/ 2.0 t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.8e-51) || !(t <= 2.1e-98)) {
		tmp = 2.0 / pow((((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))) * cbrt((tan(k) * (2.0 + pow((k / t), 2.0))))), 3.0);
	} else {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.8e-51) || !(t <= 2.1e-98)) {
		tmp = 2.0 / Math.pow((((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))))), 3.0);
	} else {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.8e-51) || !(t <= 2.1e-98))
		tmp = Float64(2.0 / (Float64(Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.8e-51], N[Not[LessEqual[t, 2.1e-98]], $MachinePrecision]], N[(2.0 / N[Power[N[(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] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e-51 or 2.09999999999999992e-98 < t

   1. Initial program 65.1%

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

      1. associate-/r* 71.9%

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

      2. add-cube-cbrt 71.8%

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

      3. *-un-lft-identity 71.8%

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

      4. times-frac 71.8%

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

      5. pow2 71.8%

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

      6. cbrt-div 71.7%

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

      7. rem-cbrt-cube 71.8%

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

      8. cbrt-div 71.7%

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

      9. rem-cbrt-cube 79.5%

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

   3. Applied egg-rr 79.5%

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

      1. /-rgt-identity 79.5%

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

      2. associate-*r/ 76.0%

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

      3. add-cube-cbrt 76.0%

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

      4. pow3 76.0%

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

      5. cbrt-prod 76.0%

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

      6. cbrt-div 78.2%

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

      7. unpow2 78.2%

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

      8. add-cbrt-cube 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. add-cube-cbrt 87.6%

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

      2. pow3 87.6%

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

   7. Applied egg-rr 94.7%

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

      1. associate-+r+ 94.7%

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

      2. metadata-eval 94.7%

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

   9. Simplified 94.7%

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

   if -1.8e-51 < t < 2.09999999999999992e-98

   1. Initial program 41.2%

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

   3. Simplified 40.1%

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

      1. associate-*r* 46.3%

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

      2. *-un-lft-identity 46.3%

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

      3. times-frac 46.3%

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

   5. Applied egg-rr 46.3%

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

      1. /-rgt-identity 46.3%

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

      2. associate-*r/ 46.3%

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

      3. times-frac 46.4%

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

      4. *-commutative 46.4%

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

      5. associate-/r* 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in t around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. *-commutative 78.8%

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

      3. times-frac 83.9%

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

   10. Simplified 83.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-51} \lor \neg \left(t \leq 2.1 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 2: 78.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < +inf.0

   1. Initial program 80.6%

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

   3. Simplified 74.1%

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

      1. associate-*r* 77.4%

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

      2. *-commutative 77.4%

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

      3. times-frac 80.6%

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

   5. Applied egg-rr 80.6%

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

      1. expm1-log1p-u 63.2%

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

      2. expm1-udef 59.0%

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

      3. times-frac 65.3%

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

   7. Applied egg-rr 65.3%

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

      1. expm1-def 70.6%

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

      2. expm1-log1p 88.1%

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

      3. associate-*r/ 88.1%

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

      4. associate-/l/ 88.2%

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

      5. associate-*l/ 88.7%

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

   9. Simplified 88.7%

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

   if +inf.0 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. associate-*r* 11.6%

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

      2. *-un-lft-identity 11.6%

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

      3. times-frac 11.5%

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

   5. Applied egg-rr 11.5%

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

      1. /-rgt-identity 11.5%

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

      2. associate-*r/ 11.6%

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

      3. times-frac 11.6%

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

      4. *-commutative 11.6%

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

      5. associate-/r* 11.6%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in t around 0 49.5%

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

      1. associate-*r/ 49.5%

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

      2. *-commutative 49.5%

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

      3. times-frac 54.8%

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

   10. Simplified 54.8%

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

4. Final simplification 78.3%

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

Alternative 3: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{+211}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}}{t_1}\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-51} \lor \neg \left(t \leq 2.1 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{t_1 \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -3.9e+211)
     (/
      (/ 2.0 (* (tan k) (* (sin k) (pow (/ (/ t (cbrt l)) (cbrt l)) 3.0))))
      t_1)
     (if (or (<= t -1.9e-51) (not (<= t 2.1e-100)))
       (/
        2.0
        (pow
         (* (/ t (pow (cbrt l) 2.0)) (cbrt (* t_1 (* (sin k) (tan k)))))
         3.0))
       (* (* (/ l (pow k 2.0)) (/ 2.0 t)) (/ (/ l (sin k)) (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -3.9e+211) {
		tmp = (2.0 / (tan(k) * (sin(k) * pow(((t / cbrt(l)) / cbrt(l)), 3.0)))) / t_1;
	} else if ((t <= -1.9e-51) || !(t <= 2.1e-100)) {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((t_1 * (sin(k) * tan(k))))), 3.0);
	} else {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	}
	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 tmp;
	if (t <= -3.9e+211) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(((t / Math.cbrt(l)) / Math.cbrt(l)), 3.0)))) / t_1;
	} else if ((t <= -1.9e-51) || !(t <= 2.1e-100)) {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((t_1 * (Math.sin(k) * Math.tan(k))))), 3.0);
	} else {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -3.9e+211)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(Float64(t / cbrt(l)) / cbrt(l)) ^ 3.0)))) / t_1);
	elseif ((t <= -1.9e-51) || !(t <= 2.1e-100))
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(t_1 * Float64(sin(k) * tan(k))))) ^ 3.0));
	else
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	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]}, If[LessEqual[t, -3.9e+211], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[Or[LessEqual[t, -1.9e-51], N[Not[LessEqual[t, 2.1e-100]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.90000000000000023e211

   1. Initial program 70.6%

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

   3. Simplified 80.7%

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

      1. add-cube-cbrt 80.7%

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

      2. pow2 80.7%

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

      3. cbrt-div 80.7%

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

      4. rem-cbrt-cube 80.7%

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

      5. cbrt-div 80.7%

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

      6. rem-cbrt-cube 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. add-cube-cbrt 80.7%

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

      2. pow3 80.7%

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

      3. cbrt-div 80.7%

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

      4. unpow2 80.7%

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

      5. add-cbrt-cube 86.0%

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

   7. Applied egg-rr 86.0%

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

   if -3.90000000000000023e211 < t < -1.90000000000000001e-51 or 2.10000000000000009e-100 < t

   1. Initial program 64.4%

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

      1. associate-/r* 70.8%

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

      2. add-cube-cbrt 70.7%

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

      3. *-un-lft-identity 70.7%

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

      4. times-frac 70.7%

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

      5. pow2 70.7%

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

      6. cbrt-div 70.6%

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

      7. rem-cbrt-cube 70.7%

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

      8. cbrt-div 70.6%

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

      9. rem-cbrt-cube 79.4%

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

   3. Applied egg-rr 79.4%

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

      1. distribute-lft-in 79.4%

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

      2. associate-*l* 75.0%

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

      3. frac-times 71.6%

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

      4. pow-plus 71.6%

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

      5. metadata-eval 71.6%

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

      6. *-un-lft-identity 71.6%

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

   5. Applied egg-rr 71.5%

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

      1. distribute-lft-out 71.5%

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

      2. +-commutative 71.5%

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

      3. associate-+r+ 71.5%

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

      4. metadata-eval 71.5%

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

      5. associate-*l* 71.5%

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

   7. Simplified 71.5%

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

      1. add-cube-cbrt 71.5%

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

      2. pow3 71.5%

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

      3. cbrt-prod 71.5%

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

      4. cbrt-div 75.5%

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

      5. unpow3 75.4%

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

      6. add-cbrt-cube 83.0%

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

      7. associate-/l/ 83.0%

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

      8. pow2 83.0%

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

   9. Applied egg-rr 83.0%

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

   if -1.90000000000000001e-51 < t < 2.10000000000000009e-100

   1. Initial program 41.2%

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

   3. Simplified 40.1%

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

      1. associate-*r* 46.3%

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

      2. *-un-lft-identity 46.3%

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

      3. times-frac 46.3%

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

   5. Applied egg-rr 46.3%

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

      1. /-rgt-identity 46.3%

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

      2. associate-*r/ 46.3%

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

      3. times-frac 46.4%

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

      4. *-commutative 46.4%

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

      5. associate-/r* 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in t around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. *-commutative 78.8%

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

      3. times-frac 83.9%

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

   10. Simplified 83.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+211}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-51} \lor \neg \left(t \leq 2.1 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 4: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-51} \lor \neg \left(t \leq 2.7 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2e-51) (not (<= t 2.7e-91)))
   (/
    2.0
    (*
     (* (tan k) (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0))
     (+ 1.0 (+ (pow (/ k t) 2.0) 1.0))))
   (* (* (/ l (pow k 2.0)) (/ 2.0 t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2e-51) || !(t <= 2.7e-91)) {
		tmp = 2.0 / ((tan(k) * pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0)) * (1.0 + (pow((k / t), 2.0) + 1.0)));
	} else {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2e-51) || !(t <= 2.7e-91)) {
		tmp = 2.0 / ((Math.tan(k) * Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0)) * (1.0 + (Math.pow((k / t), 2.0) + 1.0)));
	} else {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2e-51) || !(t <= 2.7e-91))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0)) * Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0))));
	else
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2e-51], N[Not[LessEqual[t, 2.7e-91]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * 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]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2e-51 or 2.6999999999999997e-91 < t

   1. Initial program 65.3%

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

      1. associate-/r* 72.2%

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

      2. add-cube-cbrt 72.1%

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

      3. *-un-lft-identity 72.1%

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

      4. times-frac 72.1%

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

      5. pow2 72.1%

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

      6. cbrt-div 72.0%

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

      7. rem-cbrt-cube 72.1%

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

      8. cbrt-div 72.0%

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

      9. rem-cbrt-cube 79.9%

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

   3. Applied egg-rr 79.9%

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

      1. /-rgt-identity 79.9%

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

      2. associate-*r/ 76.3%

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

      3. add-cube-cbrt 76.3%

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

      4. pow3 76.3%

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

      5. cbrt-prod 76.3%

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

      6. cbrt-div 78.5%

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

      7. unpow2 78.5%

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

      8. add-cbrt-cube 88.1%

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

   5. Applied egg-rr 88.1%

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

      1. expm1-log1p-u 51.9%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 41.0%

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

      3. associate-/l/ 41.0%

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

      4. pow2 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. expm1-def 51.9%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 88.2%

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

   9. Simplified 88.2%

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

   if -2e-51 < t < 2.6999999999999997e-91

   1. Initial program 41.4%

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

   3. Simplified 40.3%

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

      1. associate-*r* 46.4%

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

      2. *-un-lft-identity 46.4%

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

      3. times-frac 46.4%

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

   5. Applied egg-rr 46.4%

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

      1. /-rgt-identity 46.4%

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

      2. associate-*r/ 46.4%

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

      3. times-frac 46.4%

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

      4. *-commutative 46.4%

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

      5. associate-/r* 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in t around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. *-commutative 78.2%

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

      3. times-frac 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-51} \lor \neg \left(t \leq 2.7 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 5: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt[3]{\ell}}\\ t_2 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({t_1}^{2} \cdot \frac{t_1}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{if}\;t \leq -26000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-56}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k \cdot {t}^{3}}}{\tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cbrt l)))
        (t_2
         (/
          2.0
          (*
           (* (tan k) (* (sin k) (* (pow t_1 2.0) (/ t_1 l))))
           (+ 1.0 (+ 1.0 (* (/ k t) (/ k t))))))))
   (if (<= t -26000000000.0)
     t_2
     (if (<= t 5.4e-56)
       (* (* (/ l (pow k 2.0)) (/ 2.0 t)) (/ (/ l (sin k)) (tan k)))
       (if (<= t 5.6e+102)
         (*
          (/ (/ (* 2.0 l) (* (sin k) (pow t 3.0))) (tan k))
          (/ l (+ 2.0 (pow (/ k t) 2.0))))
         t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = t / cbrt(l);
	double t_2 = 2.0 / ((tan(k) * (sin(k) * (pow(t_1, 2.0) * (t_1 / l)))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	double tmp;
	if (t <= -26000000000.0) {
		tmp = t_2;
	} else if (t <= 5.4e-56) {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	} else if (t <= 5.6e+102) {
		tmp = (((2.0 * l) / (sin(k) * pow(t, 3.0))) / tan(k)) * (l / (2.0 + pow((k / t), 2.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.cbrt(l);
	double t_2 = 2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_1, 2.0) * (t_1 / l)))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	double tmp;
	if (t <= -26000000000.0) {
		tmp = t_2;
	} else if (t <= 5.4e-56) {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else if (t <= 5.6e+102) {
		tmp = (((2.0 * l) / (Math.sin(k) * Math.pow(t, 3.0))) / Math.tan(k)) * (l / (2.0 + Math.pow((k / t), 2.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / cbrt(l))
	t_2 = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_1 ^ 2.0) * Float64(t_1 / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t))))))
	tmp = 0.0
	if (t <= -26000000000.0)
		tmp = t_2;
	elseif (t <= 5.4e-56)
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	elseif (t <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * (t ^ 3.0))) / tan(k)) * Float64(l / Float64(2.0 + (Float64(k / t) ^ 2.0))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -26000000000.0], t$95$2, If[LessEqual[t, 5.4e-56], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+102], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.6e10 or 5.60000000000000037e102 < t

   1. Initial program 64.5%

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

      1. associate-/r* 70.0%

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

      2. add-cube-cbrt 70.0%

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

      3. *-un-lft-identity 70.0%

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

      4. times-frac 69.9%

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

      5. pow2 69.9%

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

      6. cbrt-div 69.9%

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

      7. rem-cbrt-cube 70.0%

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

      8. cbrt-div 70.0%

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

      9. rem-cbrt-cube 81.3%

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

   3. Applied egg-rr 81.3%

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

      1. unpow2 81.3%

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

   5. Applied egg-rr 81.3%

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

   if -2.6e10 < t < 5.3999999999999999e-56

   1. Initial program 44.5%

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

   3. Simplified 43.6%

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

      1. associate-*r* 50.3%

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

      2. *-un-lft-identity 50.3%

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

      3. times-frac 51.1%

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

   5. Applied egg-rr 51.1%

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

      1. /-rgt-identity 51.1%

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

      2. associate-*r/ 50.3%

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

      3. times-frac 51.1%

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

      4. *-commutative 51.1%

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

      5. associate-/r* 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in t around 0 76.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. times-frac 80.5%

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

   10. Simplified 80.5%

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

   if 5.3999999999999999e-56 < t < 5.60000000000000037e102

   1. Initial program 71.8%

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

   3. Simplified 67.9%

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

      1. associate-*r* 74.8%

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

      2. *-commutative 74.8%

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

      3. times-frac 88.8%

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

   5. Applied egg-rr 88.8%

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

      1. expm1-log1p-u 66.3%

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

      2. expm1-udef 52.5%

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

      3. times-frac 56.3%

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

   7. Applied egg-rr 56.3%

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

      1. expm1-def 77.2%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.7%

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

      4. associate-/l/ 99.8%

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

      5. associate-*l/ 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -26000000000:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-56}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k \cdot {t}^{3}}}{\tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array} \]

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{{t}^{3}}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+212}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\ell}^{2}}{k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{t}\right)}^{3}}{2 \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-51} \lor \neg \left(t \leq 8 \cdot 10^{-91}\right):\\ \;\;\;\;t_2 \cdot \frac{\ell \cdot t_1}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (pow t 3.0))) (t_2 (/ (/ l (sin k)) (tan k))))
   (if (<= t -1.8e+212)
     (* (* 0.5 (/ (pow l 2.0) k)) (/ t_1 (tan k)))
     (if (<= t -5.5e+102)
       (/ (pow (/ (cbrt (* 2.0 (pow l 2.0))) t) 3.0) (* 2.0 (pow k 2.0)))
       (if (or (<= t -2.1e-51) (not (<= t 8e-91)))
         (* t_2 (/ (* l t_1) (+ 2.0 (pow (/ k t) 2.0))))
         (* (* (/ l (pow k 2.0)) (/ 2.0 t)) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / pow(t, 3.0);
	double t_2 = (l / sin(k)) / tan(k);
	double tmp;
	if (t <= -1.8e+212) {
		tmp = (0.5 * (pow(l, 2.0) / k)) * (t_1 / tan(k));
	} else if (t <= -5.5e+102) {
		tmp = pow((cbrt((2.0 * pow(l, 2.0))) / t), 3.0) / (2.0 * pow(k, 2.0));
	} else if ((t <= -2.1e-51) || !(t <= 8e-91)) {
		tmp = t_2 * ((l * t_1) / (2.0 + pow((k / t), 2.0)));
	} else {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / Math.pow(t, 3.0);
	double t_2 = (l / Math.sin(k)) / Math.tan(k);
	double tmp;
	if (t <= -1.8e+212) {
		tmp = (0.5 * (Math.pow(l, 2.0) / k)) * (t_1 / Math.tan(k));
	} else if (t <= -5.5e+102) {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow(l, 2.0))) / t), 3.0) / (2.0 * Math.pow(k, 2.0));
	} else if ((t <= -2.1e-51) || !(t <= 8e-91)) {
		tmp = t_2 * ((l * t_1) / (2.0 + Math.pow((k / t), 2.0)));
	} else {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / (t ^ 3.0))
	t_2 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if (t <= -1.8e+212)
		tmp = Float64(Float64(0.5 * Float64((l ^ 2.0) / k)) * Float64(t_1 / tan(k)));
	elseif (t <= -5.5e+102)
		tmp = Float64((Float64(cbrt(Float64(2.0 * (l ^ 2.0))) / t) ^ 3.0) / Float64(2.0 * (k ^ 2.0)));
	elseif ((t <= -2.1e-51) || !(t <= 8e-91))
		tmp = Float64(t_2 * Float64(Float64(l * t_1) / Float64(2.0 + (Float64(k / t) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * t_2);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8e+212], N[(N[(0.5 * N[(N[Power[l, 2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.5e+102], N[(N[Power[N[(N[Power[N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -2.1e-51], N[Not[LessEqual[t, 8e-91]], $MachinePrecision]], N[(t$95$2 * N[(N[(l * t$95$1), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.8e212

   1. Initial program 70.6%

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

   3. Simplified 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-*r* 35.3%

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

      3. times-frac 70.6%

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

      4. pow2 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in k around 0 70.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\ell}^{2}}{k}\right)} \cdot \frac{\frac{2}{{t}^{3}}}{\tan k} \]

   if -1.8e212 < t < -5.49999999999999981e102

   1. Initial program 49.9%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in k around 0 49.3%

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

      1. add-cube-cbrt 49.3%

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

      2. pow2 49.3%

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

      3. cbrt-div 49.3%

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

      4. associate-*l/ 49.3%

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

      5. cbrt-div 49.3%

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

      6. pow2 49.3%

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

      7. unpow3 49.3%

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

      8. add-cbrt-cube 49.3%

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

      9. cbrt-div 49.3%

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

   6. Applied egg-rr 71.8%

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

      1. pow-plus 71.8%

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

      2. metadata-eval 71.8%

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

      3. cube-div 63.4%

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

      4. rem-cube-cbrt 63.6%

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

   8. Simplified 63.6%

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

   if -5.49999999999999981e102 < t < -2.10000000000000002e-51 or 8.00000000000000018e-91 < t

   1. Initial program 68.5%

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

   3. Simplified 64.6%

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

      1. associate-*r* 71.0%

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

      2. *-un-lft-identity 71.0%

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

      3. times-frac 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. /-rgt-identity 77.2%

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

      2. associate-*r/ 71.0%

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

      3. times-frac 77.2%

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

      4. *-commutative 77.2%

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

      5. associate-/r* 80.0%

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

   7. Simplified 80.0%

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

   if -2.10000000000000002e-51 < t < 8.00000000000000018e-91

   1. Initial program 41.4%

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

   3. Simplified 40.3%

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

      1. associate-*r* 46.4%

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

      2. *-un-lft-identity 46.4%

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

      3. times-frac 46.4%

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

   5. Applied egg-rr 46.4%

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

      1. /-rgt-identity 46.4%

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

      2. associate-*r/ 46.4%

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

      3. times-frac 46.4%

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

      4. *-commutative 46.4%

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

      5. associate-/r* 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in t around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. *-commutative 78.2%

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

      3. times-frac 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+212}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\ell}^{2}}{k}\right) \cdot \frac{\frac{2}{{t}^{3}}}{\tan k}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{t}\right)}^{3}}{2 \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-51} \lor \neg \left(t \leq 8 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 7: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -9000000000:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k \cdot {t}^{3}}}{\tan k} \cdot \frac{\ell}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -9000000000.0)
     (/
      2.0
      (*
       (+ 1.0 (+ t_1 1.0))
       (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
     (if (<= t 7e-50)
       (* (* (/ l (pow k 2.0)) (/ 2.0 t)) (/ (/ l (sin k)) (tan k)))
       (*
        (/ (/ (* 2.0 l) (* (sin k) (pow t 3.0))) (tan k))
        (/ l (+ 2.0 t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -9000000000.0) {
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= 7e-50) {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	} else {
		tmp = (((2.0 * l) / (sin(k) * pow(t, 3.0))) / tan(k)) * (l / (2.0 + 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 = (k / t) ** 2.0d0
    if (t <= (-9000000000.0d0)) then
        tmp = 2.0d0 / ((1.0d0 + (t_1 + 1.0d0)) * (tan(k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l)))))
    else if (t <= 7d-50) then
        tmp = ((l / (k ** 2.0d0)) * (2.0d0 / t)) * ((l / sin(k)) / tan(k))
    else
        tmp = (((2.0d0 * l) / (sin(k) * (t ** 3.0d0))) / tan(k)) * (l / (2.0d0 + t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -9000000000.0) {
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else if (t <= 7e-50) {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = (((2.0 * l) / (Math.sin(k) * Math.pow(t, 3.0))) / Math.tan(k)) * (l / (2.0 + t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if t <= -9000000000.0:
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (math.tan(k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l)))))
	elif t <= 7e-50:
		tmp = ((l / math.pow(k, 2.0)) * (2.0 / t)) * ((l / math.sin(k)) / math.tan(k))
	else:
		tmp = (((2.0 * l) / (math.sin(k) * math.pow(t, 3.0))) / math.tan(k)) * (l / (2.0 + t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -9000000000.0)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	elseif (t <= 7e-50)
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * (t ^ 3.0))) / tan(k)) * Float64(l / Float64(2.0 + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if (t <= -9000000000.0)
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (((t ^ 2.0) / l) * (t / l)))));
	elseif (t <= 7e-50)
		tmp = ((l / (k ^ 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	else
		tmp = (((2.0 * l) / (sin(k) * (t ^ 3.0))) / tan(k)) * (l / (2.0 + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -9000000000.0], N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-50], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9e9

   1. Initial program 61.6%

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

      1. unpow3 61.6%

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

      2. times-frac 75.8%

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

      3. pow2 75.8%

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

   3. Applied egg-rr 75.8%

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

   if -9e9 < t < 6.99999999999999993e-50

   1. Initial program 44.5%

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

   3. Simplified 43.6%

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

      1. associate-*r* 50.3%

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

      2. *-un-lft-identity 50.3%

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

      3. times-frac 51.1%

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

   5. Applied egg-rr 51.1%

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

      1. /-rgt-identity 51.1%

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

      2. associate-*r/ 50.3%

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

      3. times-frac 51.1%

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

      4. *-commutative 51.1%

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

      5. associate-/r* 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in t around 0 76.3%

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

      1. associate-*r/ 76.3%

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

      2. *-commutative 76.3%

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

      3. times-frac 80.5%

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

   10. Simplified 80.5%

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

   if 6.99999999999999993e-50 < t

   1. Initial program 70.3%

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

   3. Simplified 65.5%

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

      1. associate-*r* 70.0%

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

      2. *-commutative 70.0%

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

      3. times-frac 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. expm1-log1p-u 66.6%

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

      2. expm1-udef 61.1%

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

      3. times-frac 65.8%

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

   7. Applied egg-rr 65.8%

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

      1. expm1-def 74.1%

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

      2. expm1-log1p 83.0%

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

      3. associate-*r/ 83.2%

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

      4. associate-/l/ 83.2%

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

      5. associate-*l/ 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9000000000:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\sin k \cdot {t}^{3}}}{\tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 8: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-51} \lor \neg \left(t \leq 1.2 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.8e-51) (not (<= t 1.2e-91)))
   (*
    (/ (* l (/ 2.0 (pow t 3.0))) (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
    (/ l (tan k)))
   (* (* (/ l (pow k 2.0)) (/ 2.0 t)) (/ (/ l (sin k)) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.8e-51) || !(t <= 1.2e-91)) {
		tmp = ((l * (2.0 / pow(t, 3.0))) / (sin(k) * (2.0 + pow((k / t), 2.0)))) * (l / tan(k));
	} else {
		tmp = ((l / pow(k, 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.8d-51)) .or. (.not. (t <= 1.2d-91))) then
        tmp = ((l * (2.0d0 / (t ** 3.0d0))) / (sin(k) * (2.0d0 + ((k / t) ** 2.0d0)))) * (l / tan(k))
    else
        tmp = ((l / (k ** 2.0d0)) * (2.0d0 / t)) * ((l / sin(k)) / tan(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.8e-51) || !(t <= 1.2e-91)) {
		tmp = ((l * (2.0 / Math.pow(t, 3.0))) / (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0)))) * (l / Math.tan(k));
	} else {
		tmp = ((l / Math.pow(k, 2.0)) * (2.0 / t)) * ((l / Math.sin(k)) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.8e-51) or not (t <= 1.2e-91):
		tmp = ((l * (2.0 / math.pow(t, 3.0))) / (math.sin(k) * (2.0 + math.pow((k / t), 2.0)))) * (l / math.tan(k))
	else:
		tmp = ((l / math.pow(k, 2.0)) * (2.0 / t)) * ((l / math.sin(k)) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.8e-51) || !(t <= 1.2e-91))
		tmp = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(2.0 / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.8e-51) || ~((t <= 1.2e-91)))
		tmp = ((l * (2.0 / (t ^ 3.0))) / (sin(k) * (2.0 + ((k / t) ^ 2.0)))) * (l / tan(k));
	else
		tmp = ((l / (k ^ 2.0)) * (2.0 / t)) * ((l / sin(k)) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.8e-51], N[Not[LessEqual[t, 1.2e-91]], $MachinePrecision]], N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e-51 or 1.20000000000000005e-91 < t

   1. Initial program 65.3%

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

   3. Simplified 58.5%

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

      1. associate-*r* 64.1%

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

      2. associate-*r* 64.2%

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

      3. times-frac 76.4%

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

   5. Applied egg-rr 76.4%

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

   if -1.8e-51 < t < 1.20000000000000005e-91

   1. Initial program 41.4%

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

   3. Simplified 40.3%

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

      1. associate-*r* 46.4%

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

      2. *-un-lft-identity 46.4%

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

      3. times-frac 46.4%

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

   5. Applied egg-rr 46.4%

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

      1. /-rgt-identity 46.4%

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

      2. associate-*r/ 46.4%

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

      3. times-frac 46.4%

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

      4. *-commutative 46.4%

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

      5. associate-/r* 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in t around 0 78.2%

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

      1. associate-*r/ 78.2%

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

      2. *-commutative 78.2%

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

      3. times-frac 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-51} \lor \neg \left(t \leq 1.2 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 9: 66.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;k \leq 1.04 \cdot 10^{-20}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l (sin k)) (tan k))))
   (if (<= k 1.04e-20)
     (* t_1 (/ l (pow t 3.0)))
     (* t_1 (* 2.0 (/ l (* t (pow k 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = (l / sin(k)) / tan(k);
	double tmp;
	if (k <= 1.04e-20) {
		tmp = t_1 * (l / pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (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 / sin(k)) / tan(k)
    if (k <= 1.04d-20) then
        tmp = t_1 * (l / (t ** 3.0d0))
    else
        tmp = t_1 * (2.0d0 * (l / (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.sin(k)) / Math.tan(k);
	double tmp;
	if (k <= 1.04e-20) {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (t * Math.pow(k, 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / math.sin(k)) / math.tan(k)
	tmp = 0
	if k <= 1.04e-20:
		tmp = t_1 * (l / math.pow(t, 3.0))
	else:
		tmp = t_1 * (2.0 * (l / (t * math.pow(k, 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / sin(k)) / tan(k))
	tmp = 0.0
	if (k <= 1.04e-20)
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(l / Float64(t * (k ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / sin(k)) / tan(k);
	tmp = 0.0;
	if (k <= 1.04e-20)
		tmp = t_1 * (l / (t ^ 3.0));
	else
		tmp = t_1 * (2.0 * (l / (t * (k ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.04e-20], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(l / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.04000000000000007e-20

   1. Initial program 58.7%

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

   3. Simplified 53.4%

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

      1. associate-*r* 59.6%

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

      2. *-un-lft-identity 59.6%

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

      3. times-frac 62.4%

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

   5. Applied egg-rr 62.4%

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

      1. /-rgt-identity 62.4%

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

      2. associate-*r/ 59.6%

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

      3. times-frac 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-/r* 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in t around inf 63.2%

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

   if 1.04000000000000007e-20 < k

   1. Initial program 46.2%

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

   3. Simplified 44.4%

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

      1. associate-*r* 48.9%

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

      2. *-un-lft-identity 48.9%

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

      3. times-frac 50.6%

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

   5. Applied egg-rr 50.6%

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

      1. /-rgt-identity 50.6%

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

      2. associate-*r/ 48.9%

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

      3. times-frac 50.6%

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

      4. *-commutative 50.6%

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

      5. associate-/r* 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in t around 0 67.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.04 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \]

Alternative 10: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e+100)
   (* (/ (/ l (sin k)) (tan k)) (/ l (pow t 3.0)))
   (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t (cbrt l)) 3.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e+100) {
		tmp = ((l / sin(k)) / tan(k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow((t / cbrt(l)), 3.0) / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e+100) {
		tmp = ((l / Math.sin(k)) / Math.tan(k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t / Math.cbrt(l)), 3.0) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e+100)
		tmp = Float64(Float64(Float64(l / sin(k)) / tan(k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.3e+100], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3000000000000001e100

   1. Initial program 58.2%

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

   3. Simplified 53.5%

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

      1. associate-*r* 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. times-frac 61.9%

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

   5. Applied egg-rr 61.9%

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

      1. /-rgt-identity 61.9%

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

      2. associate-*r/ 59.4%

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

      3. times-frac 62.0%

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

      4. *-commutative 62.0%

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

      5. associate-/r* 65.1%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in t around inf 63.7%

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

   if 3.3000000000000001e100 < k

   1. Initial program 40.3%

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

      1. associate-/r* 47.6%

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

      2. add-cube-cbrt 47.5%

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

      3. *-un-lft-identity 47.5%

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

      4. times-frac 47.5%

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

      5. pow2 47.5%

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

      6. cbrt-div 47.6%

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

      7. rem-cbrt-cube 47.6%

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

      8. cbrt-div 47.6%

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

      9. rem-cbrt-cube 55.8%

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

   3. Applied egg-rr 55.8%

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

      1. distribute-lft-in 55.8%

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

      2. associate-*l* 55.7%

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

      3. frac-times 50.2%

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

      4. pow-plus 50.2%

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

      5. metadata-eval 50.2%

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

      6. *-un-lft-identity 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. distribute-lft-out 50.2%

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

      2. +-commutative 50.2%

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

      3. associate-+r+ 50.2%

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

      4. metadata-eval 50.2%

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

      5. associate-*l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in k around 0 39.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]

Alternative 11: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e+100)
   (* (/ (/ l (sin k)) (tan k)) (/ l (pow t 3.0)))
   (/ (/ (pow l 2.0) (pow k 2.0)) (pow t 3.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e+100) {
		tmp = ((l / sin(k)) / tan(k)) * (l / pow(t, 3.0));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) / pow(t, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.3d+100) then
        tmp = ((l / sin(k)) / tan(k)) * (l / (t ** 3.0d0))
    else
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e+100) {
		tmp = ((l / Math.sin(k)) / Math.tan(k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.3e+100:
		tmp = ((l / math.sin(k)) / math.tan(k)) * (l / math.pow(t, 3.0))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e+100)
		tmp = Float64(Float64(Float64(l / sin(k)) / tan(k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.3e+100)
		tmp = ((l / sin(k)) / tan(k)) * (l / (t ^ 3.0));
	else
		tmp = ((l ^ 2.0) / (k ^ 2.0)) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.3e+100], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3000000000000001e100

   1. Initial program 58.2%

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

   3. Simplified 53.5%

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

      1. associate-*r* 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. times-frac 61.9%

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

   5. Applied egg-rr 61.9%

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

      1. /-rgt-identity 61.9%

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

      2. associate-*r/ 59.4%

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

      3. times-frac 62.0%

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

      4. *-commutative 62.0%

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

      5. associate-/r* 65.1%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in t around inf 63.7%

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

   if 3.3000000000000001e100 < k

   1. Initial program 40.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in k around 0 40.3%

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

      1. associate-/r* 40.4%

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

   6. Simplified 40.4%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}\\ \end{array} \]

Alternative 12: 51.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\ell}^{2} \cdot \left({k}^{-2} \cdot {t}^{-3}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (pow l 2.0) (* (pow k -2.0) (pow t -3.0))))

C

double code(double t, double l, double k) {
	return pow(l, 2.0) * (pow(k, -2.0) * 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 ** 2.0d0) * ((k ** (-2.0d0)) * (t ** (-3.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return Math.pow(l, 2.0) * (Math.pow(k, -2.0) * Math.pow(t, -3.0));
}

Python

def code(t, l, k):
	return math.pow(l, 2.0) * (math.pow(k, -2.0) * math.pow(t, -3.0))

Julia

function code(t, l, k)
	return Float64((l ^ 2.0) * Float64((k ^ -2.0) * (t ^ -3.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l ^ 2.0) * ((k ^ -2.0) * (t ^ -3.0));
end

Wolfram

code[t_, l_, k_] := N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k, -2.0], $MachinePrecision] * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Simplified 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. Taylor expanded in k around 0 53.7%

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

   1. associate-/r* 53.7%

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

6. Simplified 53.7%

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

   1. expm1-log1p-u 37.6%

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

   2. expm1-udef 36.1%

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

   3. div-inv 35.3%

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

   4. div-inv 34.5%

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

   5. pow-flip 34.6%

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

   6. metadata-eval 34.6%

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

   7. pow-flip 34.6%

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

   8. metadata-eval 34.6%

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

8. Applied egg-rr 34.6%

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

   1. expm1-def 36.0%

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

   2. expm1-log1p 52.6%

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

   3. associate-*l* 52.6%

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

10. Simplified 52.6%

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

11. Final simplification 52.6%

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

Alternative 13: 51.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ (pow l 2.0) (pow k 2.0)) (pow t 3.0)))

C

double code(double t, double l, double k) {
	return (pow(l, 2.0) / pow(k, 2.0)) / 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 ** 2.0d0) / (k ** 2.0d0)) / (t ** 3.0d0)
end function

Java

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

Python

def code(t, l, k):
	return (math.pow(l, 2.0) / math.pow(k, 2.0)) / math.pow(t, 3.0)

Julia

function code(t, l, k)
	return Float64(Float64((l ^ 2.0) / (k ^ 2.0)) / (t ^ 3.0))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l ^ 2.0) / (k ^ 2.0)) / (t ^ 3.0);
end

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Simplified 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. Taylor expanded in k around 0 53.7%

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

   1. associate-/r* 53.7%

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

6. Simplified 53.7%

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

7. Final simplification 53.7%

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

Alternative 14: 51.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0))))

C

double code(double t, double l, double k) {
	return pow(l, 2.0) / (pow(k, 2.0) * 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 ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Simplified 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. Taylor expanded in k around 0 53.7%

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

5. Final simplification 53.7%

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

Alternative 15: 51.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* (* l l) (* 2.0 (pow t -3.0))) (* 2.0 (pow k 2.0))))

C

double code(double t, double l, double k) {
	return ((l * l) * (2.0 * pow(t, -3.0))) / (2.0 * pow(k, 2.0));
}

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 * l) * (2.0d0 * (t ** (-3.0d0)))) / (2.0d0 * (k ** 2.0d0))
end function

Java

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

Python

def code(t, l, k):
	return ((l * l) * (2.0 * math.pow(t, -3.0))) / (2.0 * math.pow(k, 2.0))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * l) * Float64(2.0 * (t ^ -3.0))) / Float64(2.0 * (k ^ 2.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * l) * (2.0 * (t ^ -3.0))) / (2.0 * (k ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Simplified 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. Taylor expanded in k around 0 51.9%

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

   1. div-inv 51.9%

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

   2. pow-flip 51.9%

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

   3. metadata-eval 51.9%

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

6. Applied egg-rr 51.9%

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

7. Final simplification 51.9%

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

Reproduce

herbie shell --seed 2023300 
(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))))