Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 81.3%

Time: 21.0s

Alternatives: 19

Speedup: 3.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}}\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        4e+146)
     (/
      (pow (/ (cbrt (/ 2.0 (tan k))) (/ t (cbrt (/ (* l l) (sin k))))) 3.0)
      (+ 2.0 t_1))
     (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* (pow (sin k) 2.0) (/ t l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / cbrt(((l * l) / sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (pow(sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / Math.cbrt(((l * l) / Math.sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 4e+146)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / cbrt(Float64(Float64(l * l) / sin(k))))) ^ 3.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64((sin(k) ^ 2.0) * Float64(t / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+146], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 3.99999999999999973e146

   1. Initial program 76.3%

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

      1. associate-/r* 75.9%

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

      2. associate-*l* 66.4%

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

      3. sqr-neg 66.4%

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

      4. associate-*l* 75.9%

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

      5. *-commutative 75.9%

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

      6. sqr-neg 75.9%

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

      7. associate-/r* 76.0%

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

   3. Simplified 77.8%

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

      1. add-cube-cbrt 77.6%

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

      2. pow3 77.6%

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

      3. cbrt-div 77.6%

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

      4. cbrt-div 77.6%

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

      5. rem-cbrt-cube 86.5%

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

   5. Applied egg-rr 86.5%

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

   if 3.99999999999999973e146 < (/.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 14.3%

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

   2. Taylor expanded in t around 0 38.7%

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

      1. times-frac 39.0%

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

      2. unpow2 39.0%

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

      3. unpow2 39.0%

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

      4. times-frac 60.7%

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

   4. Simplified 60.7%

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

      1. associate-*l/ 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. frac-times 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in k around inf 38.7%

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

       1. times-frac 39.0%

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

       2. associate-*l/ 39.0%

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

       3. unpow2 39.0%

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

       4. associate-/r* 55.7%

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

       5. associate-*r/ 60.6%

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

       6. associate-*r/ 61.5%

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

       7. unpow2 61.5%

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

       8. associate-/r* 61.5%

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

       9. unpow2 61.5%

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

       10. associate-*l/ 61.7%

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

       11. unpow2 61.7%

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

       12. times-frac 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 2: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        4e+146)
     (/
      (* (* l l) (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (tan k))) 3.0)))
      (+ 2.0 t_1))
     (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* (pow (sin k) 2.0) (/ t l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = ((l * l) * (2.0 / pow(((t * cbrt(sin(k))) * cbrt(tan(k))), 3.0))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (pow(sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = ((l * l) * (2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt(Math.tan(k))), 3.0))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 4e+146)
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(tan(k))) ^ 3.0))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64((sin(k) ^ 2.0) * Float64(t / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+146], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 3.99999999999999973e146

   1. Initial program 76.3%

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

      1. associate-/r* 75.9%

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

      2. associate-*l* 66.4%

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

      3. sqr-neg 66.4%

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

      4. associate-*l* 75.9%

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

      5. *-commutative 75.9%

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

      6. sqr-neg 75.9%

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

      7. associate-*l/ 77.9%

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

      8. associate-*r/ 75.5%

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

      9. associate-/r/ 75.5%

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

   3. Simplified 75.5%

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

      1. add-cube-cbrt 75.5%

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

      2. pow3 75.5%

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

      3. *-commutative 75.5%

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

      4. cbrt-prod 75.4%

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

      5. cbrt-prod 75.3%

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

      6. rem-cbrt-cube 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 3.99999999999999973e146 < (/.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 14.3%

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

   2. Taylor expanded in t around 0 38.7%

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

      1. times-frac 39.0%

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

      2. unpow2 39.0%

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

      3. unpow2 39.0%

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

      4. times-frac 60.7%

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

   4. Simplified 60.7%

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

      1. associate-*l/ 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. frac-times 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in k around inf 38.7%

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

       1. times-frac 39.0%

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

       2. associate-*l/ 39.0%

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

       3. unpow2 39.0%

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

       4. associate-/r* 55.7%

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

       5. associate-*r/ 60.6%

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

       6. associate-*r/ 61.5%

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

       7. unpow2 61.5%

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

       8. associate-/r* 61.5%

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

       9. unpow2 61.5%

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

       10. associate-*l/ 61.7%

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

       11. unpow2 61.7%

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

       12. times-frac 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 3: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        4e+146)
     (/ 2.0 (* t_1 (* (tan k) (pow (/ t (cbrt (/ (* l l) (sin k)))) 3.0))))
     (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* (pow (sin k) 2.0) (/ t l)))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 4e+146) {
		tmp = 2.0 / (t_1 * (tan(k) * pow((t / cbrt(((l * l) / sin(k)))), 3.0)));
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (pow(sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1)) <= 4e+146) {
		tmp = 2.0 / (t_1 * (Math.tan(k) * Math.pow((t / Math.cbrt(((l * l) / Math.sin(k)))), 3.0)));
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1)) <= 4e+146)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * (Float64(t / cbrt(Float64(Float64(l * l) / sin(k)))) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64((sin(k) ^ 2.0) * Float64(t / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision]), $MachinePrecision], 4e+146], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 3.99999999999999973e146

   1. Initial program 76.3%

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

      1. associate-/r/ 78.1%

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

      2. add-cube-cbrt 77.9%

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

      3. pow3 77.9%

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

      4. cbrt-div 78.0%

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

      5. rem-cbrt-cube 82.5%

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

   3. Applied egg-rr 82.5%

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

   if 3.99999999999999973e146 < (/.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 14.3%

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

   2. Taylor expanded in t around 0 38.7%

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

      1. times-frac 39.0%

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

      2. unpow2 39.0%

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

      3. unpow2 39.0%

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

      4. times-frac 60.7%

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

   4. Simplified 60.7%

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

      1. associate-*l/ 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. frac-times 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in k around inf 38.7%

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

       1. times-frac 39.0%

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

       2. associate-*l/ 39.0%

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

       3. unpow2 39.0%

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

       4. associate-/r* 55.7%

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

       5. associate-*r/ 60.6%

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

       6. associate-*r/ 61.5%

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

       7. unpow2 61.5%

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

       8. associate-/r* 61.5%

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

       9. unpow2 61.5%

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

       10. associate-*l/ 61.7%

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

       11. unpow2 61.7%

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

       12. times-frac 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 4: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        4e+146)
     (/
      (/ (/ (* 2.0 (* l l)) (tan k)) (pow (* t (cbrt (sin k))) 3.0))
      (+ 2.0 t_1))
     (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* (pow (sin k) 2.0) (/ t l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = (((2.0 * (l * l)) / tan(k)) / pow((t * cbrt(sin(k))), 3.0)) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (pow(sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = (((2.0 * (l * l)) / Math.tan(k)) / Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 4e+146)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(l * l)) / tan(k)) / (Float64(t * cbrt(sin(k))) ^ 3.0)) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64((sin(k) ^ 2.0) * Float64(t / l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+146], N[(N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 3.99999999999999973e146

   1. Initial program 76.3%

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

      1. associate-/r* 75.9%

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

      2. associate-*l* 66.4%

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

      3. sqr-neg 66.4%

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

      4. associate-*l* 75.9%

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

      5. *-commutative 75.9%

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

      6. sqr-neg 75.9%

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

      7. associate-*l/ 77.9%

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

      8. associate-*r/ 75.5%

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

      9. associate-/r/ 75.5%

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

   3. Simplified 75.5%

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

      1. add-cube-cbrt 75.4%

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

      2. pow3 75.4%

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

      3. cbrt-prod 75.4%

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

      4. rem-cbrt-cube 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. associate-*l/ 78.6%

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

   7. Applied egg-rr 78.6%

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

      1. associate-/r* 80.7%

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

   9. Simplified 80.7%

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

   if 3.99999999999999973e146 < (/.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 14.3%

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

   2. Taylor expanded in t around 0 38.7%

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

      1. times-frac 39.0%

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

      2. unpow2 39.0%

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

      3. unpow2 39.0%

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

      4. times-frac 60.7%

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

   4. Simplified 60.7%

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

      1. associate-*l/ 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. frac-times 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in k around inf 38.7%

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

       1. times-frac 39.0%

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

       2. associate-*l/ 39.0%

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

       3. unpow2 39.0%

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

       4. associate-/r* 55.7%

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

       5. associate-*r/ 60.6%

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

       6. associate-*r/ 61.5%

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

       7. unpow2 61.5%

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

       8. associate-/r* 61.5%

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

       9. unpow2 61.5%

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

       10. associate-*l/ 61.7%

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

       11. unpow2 61.7%

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

       12. times-frac 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 5: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        4e+146)
     (/ (/ (/ 2.0 (tan k)) (/ (pow t 3.0) (/ l (/ (sin k) l)))) (+ 2.0 t_1))
     (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* (pow (sin k) 2.0) (/ t l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = ((2.0 / tan(k)) / (pow(t, 3.0) / (l / (sin(k) / l)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (pow(sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + t_1)))) <= 4d+146) then
        tmp = ((2.0d0 / tan(k)) / ((t ** 3.0d0) / (l / (sin(k) / l)))) / (2.0d0 + t_1)
    else
        tmp = 2.0d0 / (((k / l) * (k / cos(k))) * ((sin(k) ** 2.0d0) * (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146) {
		tmp = ((2.0 / Math.tan(k)) / (Math.pow(t, 3.0) / (l / (Math.sin(k) / l)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146:
		tmp = ((2.0 / math.tan(k)) / (math.pow(t, 3.0) / (l / (math.sin(k) / l)))) / (2.0 + t_1)
	else:
		tmp = 2.0 / (((k / l) * (k / math.cos(k))) * (math.pow(math.sin(k), 2.0) * (t / l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 4e+146)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64((t ^ 3.0) / Float64(l / Float64(sin(k) / l)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64((sin(k) ^ 2.0) * Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 4e+146)
		tmp = ((2.0 / tan(k)) / ((t ^ 3.0) / (l / (sin(k) / l)))) / (2.0 + t_1);
	else
		tmp = 2.0 / (((k / l) * (k / cos(k))) * ((sin(k) ^ 2.0) * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+146], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 3.99999999999999973e146

   1. Initial program 76.3%

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

      1. associate-/r* 75.9%

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

      2. associate-*l* 66.4%

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

      3. sqr-neg 66.4%

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

      4. associate-*l* 75.9%

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

      5. *-commutative 75.9%

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

      6. sqr-neg 75.9%

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

      7. associate-/r* 76.0%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in l around 0 77.8%

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

      1. unpow2 77.8%

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

      2. associate-/l* 80.6%

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

   6. Simplified 80.6%

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

   if 3.99999999999999973e146 < (/.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 14.3%

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

   2. Taylor expanded in t around 0 38.7%

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

      1. times-frac 39.0%

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

      2. unpow2 39.0%

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

      3. unpow2 39.0%

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

      4. times-frac 60.7%

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

   4. Simplified 60.7%

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

      1. associate-*l/ 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. frac-times 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in k around inf 38.7%

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

       1. times-frac 39.0%

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

       2. associate-*l/ 39.0%

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

       3. unpow2 39.0%

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

       4. associate-/r* 55.7%

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

       5. associate-*r/ 60.6%

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

       6. associate-*r/ 61.5%

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

       7. unpow2 61.5%

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

       8. associate-/r* 61.5%

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

       9. unpow2 61.5%

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

       10. associate-*l/ 61.7%

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

       11. unpow2 61.7%

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

       12. times-frac 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 6: 70.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+227} \lor \neg \left(k \leq 9.6 \cdot 10^{+277}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_1 \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 2.4e-68)
     (* (/ l k) (/ (/ l k) (pow t 3.0)))
     (if (or (<= k 1.6e+227) (not (<= k 9.6e+277)))
       (/ 2.0 (/ (* k (* t_1 (* k (/ t l)))) (* l (cos k))))
       (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* t_1 (/ t l))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if ((k <= 1.6e+227) || !(k <= 9.6e+277)) {
		tmp = 2.0 / ((k * (t_1 * (k * (t / l)))) / (l * cos(k)));
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (t_1 * (t / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 2.4d-68) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if ((k <= 1.6d+227) .or. (.not. (k <= 9.6d+277))) then
        tmp = 2.0d0 / ((k * (t_1 * (k * (t / l)))) / (l * cos(k)))
    else
        tmp = 2.0d0 / (((k / l) * (k / cos(k))) * (t_1 * (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if ((k <= 1.6e+227) || !(k <= 9.6e+277)) {
		tmp = 2.0 / ((k * (t_1 * (k * (t / l)))) / (l * Math.cos(k)));
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (t_1 * (t / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 2.4e-68:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif (k <= 1.6e+227) or not (k <= 9.6e+277):
		tmp = 2.0 / ((k * (t_1 * (k * (t / l)))) / (l * math.cos(k)))
	else:
		tmp = 2.0 / (((k / l) * (k / math.cos(k))) * (t_1 * (t / l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 2.4e-68)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif ((k <= 1.6e+227) || !(k <= 9.6e+277))
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_1 * Float64(k * Float64(t / l)))) / Float64(l * cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64(t_1 * Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 2.4e-68)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif ((k <= 1.6e+227) || ~((k <= 9.6e+277)))
		tmp = 2.0 / ((k * (t_1 * (k * (t / l)))) / (l * cos(k)));
	else
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (t_1 * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 2.4e-68], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[k, 1.6e+227], N[Not[LessEqual[k, 9.6e+277]], $MachinePrecision]], N[(2.0 / N[(N[(k * N[(t$95$1 * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.39999999999999991e-68

   1. Initial program 57.4%

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

      1. associate-/r* 57.1%

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

      2. associate-*l* 49.1%

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

      3. sqr-neg 49.1%

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

      4. associate-*l* 57.1%

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

      5. *-commutative 57.1%

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

      6. sqr-neg 57.1%

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

      7. associate-*l/ 58.7%

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

      8. associate-*r/ 56.8%

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

      9. associate-/r/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. unpow2 49.0%

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

      2. *-commutative 49.0%

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

      3. unpow2 49.0%

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

   6. Simplified 49.0%

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

      1. add-cbrt-cube 48.5%

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

      2. times-frac 48.3%

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

      3. times-frac 48.3%

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

      4. times-frac 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. associate-*l* 52.6%

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

      2. cube-unmult 52.6%

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

      3. *-commutative 52.6%

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

      4. unpow2 52.6%

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

      5. times-frac 48.5%

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

      6. unpow2 48.5%

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

      7. associate-*l* 56.4%

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

      8. times-frac 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in l around 0 49.0%

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

       1. unpow2 49.0%

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

       2. unpow2 49.0%

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

       3. associate-*r* 58.0%

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

       4. times-frac 66.0%

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

       5. associate-/r* 67.5%

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

   13. Simplified 67.5%

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

   if 2.39999999999999991e-68 < k < 1.59999999999999994e227 or 9.59999999999999971e277 < k

   1. Initial program 37.0%

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

   2. Taylor expanded in t around 0 50.0%

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

      1. times-frac 48.2%

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

      2. unpow2 48.2%

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

      3. unpow2 48.2%

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

      4. times-frac 64.6%

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

   4. Simplified 64.6%

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

      1. associate-*l/ 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. frac-times 68.9%

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

   8. Applied egg-rr 68.9%

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

   9. Taylor expanded in k around inf 66.1%

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

       1. associate-*r/ 65.1%

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

       2. unpow2 65.1%

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

       3. associate-*r/ 68.9%

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

       4. associate-*l* 75.0%

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

       5. *-commutative 75.0%

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

       6. associate-*r* 74.9%

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

   11. Simplified 74.9%

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

   if 1.59999999999999994e227 < k < 9.59999999999999971e277

   1. Initial program 23.1%

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

   2. Taylor expanded in t around 0 38.5%

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

      1. times-frac 38.5%

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

      2. unpow2 38.5%

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

      3. unpow2 38.5%

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

      4. times-frac 46.6%

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

   4. Simplified 46.6%

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

      1. associate-*l/ 46.6%

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

   6. Applied egg-rr 46.6%

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

      1. frac-times 47.8%

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

   8. Applied egg-rr 47.8%

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

   9. Taylor expanded in k around inf 38.5%

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

       1. times-frac 38.5%

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

       2. associate-*l/ 38.5%

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

       3. unpow2 38.5%

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

       4. associate-/r* 46.6%

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

       5. associate-*r/ 46.6%

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

       6. associate-*r/ 47.8%

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

       7. unpow2 47.8%

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

       8. associate-/r* 47.8%

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

       9. unpow2 47.8%

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

       10. associate-*l/ 47.8%

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

       11. unpow2 47.8%

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

       12. times-frac 84.7%

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

   11. Simplified 84.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+227} \lor \neg \left(k \leq 9.6 \cdot 10^{+277}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \left({\sin k}^{2} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 7: 67.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.7 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-83)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 1.8e-10)
     (/ 2.0 (* (/ (* k k) (cos k)) (* (/ t l) (/ (* k k) l))))
     (if (<= k 5.7e+150)
       (* 2.0 (* (/ (cos k) (* k k)) (/ (* l l) (* t (pow (sin k) 2.0)))))
       (* 2.0 (* l (/ l (* t (pow k 4.0)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-83) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 1.8e-10) {
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * ((k * k) / l)));
	} else if (k <= 5.7e+150) {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 * (l * (l / (t * pow(k, 4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.5d-83) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 1.8d-10) then
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((t / l) * ((k * k) / l)))
    else if (k <= 5.7d+150) then
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l * l) / (t * (sin(k) ** 2.0d0))))
    else
        tmp = 2.0d0 * (l * (l / (t * (k ** 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-83) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 1.8e-10) {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((t / l) * ((k * k) / l)));
	} else if (k <= 5.7e+150) {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l * l) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = 2.0 * (l * (l / (t * Math.pow(k, 4.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.5e-83:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 1.8e-10:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((t / l) * ((k * k) / l)))
	elif k <= 5.7e+150:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l * l) / (t * math.pow(math.sin(k), 2.0))))
	else:
		tmp = 2.0 * (l * (l / (t * math.pow(k, 4.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-83)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 1.8e-10)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(t / l) * Float64(Float64(k * k) / l))));
	elseif (k <= 5.7e+150)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l * l) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-83)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 1.8e-10)
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * ((k * k) / l)));
	elseif (k <= 5.7e+150)
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * (sin(k) ^ 2.0))));
	else
		tmp = 2.0 * (l * (l / (t * (k ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.5e-83], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e-10], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.7e+150], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 2.5e-83

   1. Initial program 57.5%

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

      1. associate-/r* 57.2%

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

      2. associate-*l* 49.1%

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

      3. sqr-neg 49.1%

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

      4. associate-*l* 57.2%

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

      5. *-commutative 57.2%

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

      6. sqr-neg 57.2%

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

      7. associate-*l/ 58.8%

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

      8. associate-*r/ 56.9%

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

      9. associate-/r/ 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. unpow2 48.4%

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

      2. *-commutative 48.4%

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

      3. unpow2 48.4%

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

   6. Simplified 48.4%

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

      1. add-cbrt-cube 47.9%

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

      2. times-frac 47.8%

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

      3. times-frac 47.8%

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

      4. times-frac 52.1%

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

   8. Applied egg-rr 52.1%

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

      1. associate-*l* 52.1%

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

      2. cube-unmult 52.1%

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

      3. *-commutative 52.1%

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

      4. unpow2 52.1%

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

      5. times-frac 47.9%

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

      6. unpow2 47.9%

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

      7. associate-*l* 56.0%

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

      8. times-frac 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in l around 0 48.4%

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

       1. unpow2 48.4%

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

       2. unpow2 48.4%

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

       3. associate-*r* 57.5%

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

       4. times-frac 65.6%

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

       5. associate-/r* 67.1%

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

   13. Simplified 67.1%

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

   if 2.5e-83 < k < 1.8e-10

   1. Initial program 33.6%

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

   2. Taylor expanded in t around 0 32.9%

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

      1. times-frac 37.4%

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

      2. unpow2 37.4%

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

      3. unpow2 37.4%

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

      4. times-frac 65.7%

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

   4. Simplified 65.7%

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

   5. Taylor expanded in k around 0 37.4%

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

      1. unpow2 37.4%

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

      2. times-frac 65.7%

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

      3. unpow2 65.7%

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

   7. Simplified 65.7%

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

   if 1.8e-10 < k < 5.7000000000000002e150

   1. Initial program 44.0%

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

      1. associate-/r* 43.9%

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

      2. associate-*l* 43.9%

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

      3. sqr-neg 43.9%

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

      4. associate-*l* 43.9%

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

      5. *-commutative 43.9%

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

      6. sqr-neg 43.9%

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

      7. associate-*l/ 44.0%

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

      8. associate-*r/ 43.9%

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

      9. associate-/r/ 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in k around inf 67.7%

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

      1. times-frac 61.4%

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

      2. unpow2 61.4%

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

      3. unpow2 61.4%

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

      4. *-commutative 61.4%

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

   6. Simplified 61.4%

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

   if 5.7000000000000002e150 < k

   1. Initial program 28.0%

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

   2. Taylor expanded in t around 0 39.5%

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

      1. times-frac 39.4%

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

      2. unpow2 39.4%

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

      3. unpow2 39.4%

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

      4. times-frac 56.4%

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

   4. Simplified 56.4%

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

   5. Taylor expanded in k around 0 39.5%

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

      1. unpow2 39.5%

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

   7. Simplified 39.5%

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

   8. Taylor expanded in l around 0 39.5%

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

      1. *-rgt-identity 39.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{4} \cdot t} \]

      2. *-commutative 39.5%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot 1}{\color{blue}{t \cdot {k}^{4}}} \]

      3. associate-*r/ 39.5%

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

      4. unpow2 39.5%

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

      5. associate-*l* 57.4%

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

      6. associate-*r/ 57.4%

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

      7. *-rgt-identity 57.4%

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

      8. *-commutative 57.4%

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

   10. Simplified 57.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.7 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \]

Alternative 8: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.4e-83)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* (/ (cos k) (* k k)) (/ (* l (/ l t)) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.4e-83) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * (l / t)) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.4d-83) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l * (l / t)) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.4e-83) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l * (l / t)) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.4e-83:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l * (l / t)) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.4e-83)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l * Float64(l / t)) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.4e-83)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * (l / t)) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.4e-83], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3999999999999998e-83

   1. Initial program 57.5%

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

      1. associate-/r* 57.2%

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

      2. associate-*l* 49.1%

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

      3. sqr-neg 49.1%

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

      4. associate-*l* 57.2%

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

      5. *-commutative 57.2%

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

      6. sqr-neg 57.2%

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

      7. associate-*l/ 58.8%

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

      8. associate-*r/ 56.9%

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

      9. associate-/r/ 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. unpow2 48.4%

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

      2. *-commutative 48.4%

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

      3. unpow2 48.4%

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

   6. Simplified 48.4%

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

      1. add-cbrt-cube 47.9%

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

      2. times-frac 47.8%

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

      3. times-frac 47.8%

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

      4. times-frac 52.1%

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

   8. Applied egg-rr 52.1%

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

      1. associate-*l* 52.1%

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

      2. cube-unmult 52.1%

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

      3. *-commutative 52.1%

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

      4. unpow2 52.1%

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

      5. times-frac 47.9%

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

      6. unpow2 47.9%

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

      7. associate-*l* 56.0%

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

      8. times-frac 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in l around 0 48.4%

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

       1. unpow2 48.4%

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

       2. unpow2 48.4%

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

       3. associate-*r* 57.5%

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

       4. times-frac 65.6%

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

       5. associate-/r* 67.1%

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

   13. Simplified 67.1%

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

   if 3.3999999999999998e-83 < k

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. associate-*l* 35.0%

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

      3. sqr-neg 35.0%

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

      4. associate-*l* 35.0%

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

      5. *-commutative 35.0%

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

      6. sqr-neg 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-*r/ 34.9%

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

      9. associate-/r/ 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in k around inf 49.3%

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

      1. times-frac 47.5%

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

      2. unpow2 47.5%

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

      3. unpow2 47.5%

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

      4. *-commutative 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in l around 0 47.5%

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

      1. unpow2 47.5%

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

      2. times-frac 62.1%

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

      3. *-commutative 62.1%

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

      4. associate-*r/ 62.2%

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

   9. Simplified 62.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 9: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \cos k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.4e-68)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* (/ 2.0 (* (* (pow (sin k) 2.0) (/ t l)) (* k k))) (* l (cos k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = (2.0 / ((pow(sin(k), 2.0) * (t / l)) * (k * k))) * (l * cos(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 (k <= 2.4d-68) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = (2.0d0 / (((sin(k) ** 2.0d0) * (t / l)) * (k * k))) * (l * cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = (2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / l)) * (k * k))) * (l * Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.4e-68:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = (2.0 / ((math.pow(math.sin(k), 2.0) * (t / l)) * (k * k))) * (l * math.cos(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.4e-68)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / l)) * Float64(k * k))) * Float64(l * cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.4e-68)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = (2.0 / (((sin(k) ^ 2.0) * (t / l)) * (k * k))) * (l * cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.4e-68], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999991e-68

   1. Initial program 57.4%

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

      1. associate-/r* 57.1%

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

      2. associate-*l* 49.1%

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

      3. sqr-neg 49.1%

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

      4. associate-*l* 57.1%

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

      5. *-commutative 57.1%

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

      6. sqr-neg 57.1%

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

      7. associate-*l/ 58.7%

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

      8. associate-*r/ 56.8%

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

      9. associate-/r/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. unpow2 49.0%

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

      2. *-commutative 49.0%

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

      3. unpow2 49.0%

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

   6. Simplified 49.0%

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

      1. add-cbrt-cube 48.5%

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

      2. times-frac 48.3%

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

      3. times-frac 48.3%

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

      4. times-frac 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. associate-*l* 52.6%

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

      2. cube-unmult 52.6%

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

      3. *-commutative 52.6%

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

      4. unpow2 52.6%

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

      5. times-frac 48.5%

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

      6. unpow2 48.5%

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

      7. associate-*l* 56.4%

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

      8. times-frac 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in l around 0 49.0%

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

       1. unpow2 49.0%

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

       2. unpow2 49.0%

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

       3. associate-*r* 58.0%

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

       4. times-frac 66.0%

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

       5. associate-/r* 67.5%

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

   13. Simplified 67.5%

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

   if 2.39999999999999991e-68 < k

   1. Initial program 34.6%

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

   2. Taylor expanded in t around 0 48.0%

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

      1. times-frac 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

      4. times-frac 61.6%

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

   4. Simplified 61.6%

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

      1. associate-*l/ 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. frac-times 65.3%

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

   8. Applied egg-rr 65.3%

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

      1. associate-/r/ 65.3%

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

      2. *-commutative 65.3%

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

   10. Applied egg-rr 65.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \cos k\right)\\ \end{array} \]

Alternative 10: 69.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.4e-68)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (* (/ k l) (/ k (cos k))) (* (pow (sin k) 2.0) (/ t l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k / l) * (k / cos(k))) * (pow(sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.4d-68) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((k / l) * (k / cos(k))) * ((sin(k) ** 2.0d0) * (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k / l) * (k / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) * (t / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.4e-68:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (((k / l) * (k / math.cos(k))) * (math.pow(math.sin(k), 2.0) * (t / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.4e-68)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / cos(k))) * Float64((sin(k) ^ 2.0) * Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.4e-68)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((k / l) * (k / cos(k))) * ((sin(k) ^ 2.0) * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.4e-68], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999991e-68

   1. Initial program 57.4%

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

      1. associate-/r* 57.1%

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

      2. associate-*l* 49.1%

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

      3. sqr-neg 49.1%

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

      4. associate-*l* 57.1%

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

      5. *-commutative 57.1%

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

      6. sqr-neg 57.1%

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

      7. associate-*l/ 58.7%

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

      8. associate-*r/ 56.8%

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

      9. associate-/r/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. unpow2 49.0%

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

      2. *-commutative 49.0%

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

      3. unpow2 49.0%

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

   6. Simplified 49.0%

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

      1. add-cbrt-cube 48.5%

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

      2. times-frac 48.3%

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

      3. times-frac 48.3%

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

      4. times-frac 52.6%

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

   8. Applied egg-rr 52.6%

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

      1. associate-*l* 52.6%

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

      2. cube-unmult 52.6%

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

      3. *-commutative 52.6%

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

      4. unpow2 52.6%

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

      5. times-frac 48.5%

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

      6. unpow2 48.5%

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

      7. associate-*l* 56.4%

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

      8. times-frac 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in l around 0 49.0%

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

       1. unpow2 49.0%

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

       2. unpow2 49.0%

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

       3. associate-*r* 58.0%

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

       4. times-frac 66.0%

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

       5. associate-/r* 67.5%

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

   13. Simplified 67.5%

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

   if 2.39999999999999991e-68 < k

   1. Initial program 34.6%

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

   2. Taylor expanded in t around 0 48.0%

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

      1. times-frac 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

      4. times-frac 61.6%

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

   4. Simplified 61.6%

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

      1. associate-*l/ 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. frac-times 65.3%

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

   8. Applied egg-rr 65.3%

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

   9. Taylor expanded in k around inf 48.0%

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

       1. times-frac 46.5%

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

       2. associate-*l/ 46.6%

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

       3. unpow2 46.6%

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

       4. associate-/r* 58.5%

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

       5. associate-*r/ 61.6%

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

       6. associate-*r/ 65.3%

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

       7. unpow2 65.3%

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

       8. associate-/r* 65.3%

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

       9. unpow2 65.3%

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

       10. associate-*l/ 61.6%

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

       11. unpow2 61.6%

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

       12. times-frac 70.5%

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

   11. Simplified 70.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 11: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{t}, 0.3333333333333333, \frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -7.8e-63)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (if (<= t 3.4e-74)
     (*
      2.0
      (*
       (/ (cos k) (* k k))
       (fma (* l (/ l t)) 0.3333333333333333 (* (/ l t) (/ l (* k k))))))
     (if (<= t 1.35e+97)
       (* (/ l k) (/ (/ l k) (pow t 3.0)))
       (/ 2.0 (/ (* (* k k) (* (/ t l) (* k k))) (* l (cos k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -7.8e-63) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else if (t <= 3.4e-74) {
		tmp = 2.0 * ((cos(k) / (k * k)) * fma((l * (l / t)), 0.3333333333333333, ((l / t) * (l / (k * k)))));
	} else if (t <= 1.35e+97) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k * k) * ((t / l) * (k * k))) / (l * cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -7.8e-63)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	elseif (t <= 3.4e-74)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * fma(Float64(l * Float64(l / t)), 0.3333333333333333, Float64(Float64(l / t) * Float64(l / Float64(k * k))))));
	elseif (t <= 1.35e+97)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k * k))) / Float64(l * cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -7.8e-63], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-74], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+97], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.80000000000000044e-63

   1. Initial program 66.8%

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

      1. associate-/r* 66.8%

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

      2. associate-*l* 55.4%

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

      3. sqr-neg 55.4%

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

      4. associate-*l* 66.8%

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

      5. *-commutative 66.8%

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

      6. sqr-neg 66.8%

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

      7. associate-*l/ 66.8%

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

      8. associate-*r/ 64.9%

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

      9. associate-/r/ 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in k around 0 51.6%

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

      1. unpow2 51.6%

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

      2. *-commutative 51.6%

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

      3. unpow2 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in l around 0 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. associate-*l* 62.0%

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

      4. times-frac 68.7%

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

   9. Simplified 68.7%

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

   if -7.80000000000000044e-63 < t < 3.4000000000000001e-74

   1. Initial program 35.2%

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

      1. associate-/r* 35.1%

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

      2. associate-*l* 35.1%

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

      3. sqr-neg 35.1%

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

      4. associate-*l* 35.1%

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

      5. *-commutative 35.1%

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

      6. sqr-neg 35.1%

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

      7. associate-*l/ 35.1%

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

      8. associate-*r/ 35.1%

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

      9. associate-/r/ 35.1%

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

   3. Simplified 35.1%

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

   4. Taylor expanded in k around inf 61.4%

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

      1. times-frac 62.5%

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

      2. unpow2 62.5%

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

      3. unpow2 62.5%

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

      4. *-commutative 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in k around 0 51.7%

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

      1. +-commutative 51.7%

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

      2. *-commutative 51.7%

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

      3. fma-def 51.7%

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

      4. unpow2 51.7%

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

      5. associate-*l/ 52.0%

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

      6. unpow2 52.0%

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

      7. times-frac 70.4%

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

      8. unpow2 70.4%

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

   9. Simplified 70.4%

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

   if 3.4000000000000001e-74 < t < 1.34999999999999997e97

   1. Initial program 54.6%

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

      1. associate-/r* 53.0%

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

      2. associate-*l* 46.0%

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

      3. sqr-neg 46.0%

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

      4. associate-*l* 53.0%

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

      5. *-commutative 53.0%

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

      6. sqr-neg 53.0%

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

      7. associate-*l/ 62.5%

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

      8. associate-*r/ 57.2%

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

      9. associate-/r/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in k around 0 38.0%

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

      1. unpow2 38.0%

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

      2. *-commutative 38.0%

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

      3. unpow2 38.0%

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

   6. Simplified 38.0%

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

      1. add-cbrt-cube 34.8%

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

      2. times-frac 34.6%

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

      3. times-frac 34.6%

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

      4. times-frac 38.6%

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

   8. Applied egg-rr 38.6%

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

      1. associate-*l* 38.6%

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

      2. cube-unmult 38.6%

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

      3. *-commutative 38.6%

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

      4. unpow2 38.6%

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

      5. times-frac 34.8%

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

      6. unpow2 34.8%

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

      7. associate-*l* 44.6%

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

      8. times-frac 48.5%

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

   10. Simplified 48.5%

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

   11. Taylor expanded in l around 0 38.0%

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

       1. unpow2 38.0%

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

       2. unpow2 38.0%

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

       3. associate-*r* 50.9%

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

       4. times-frac 67.7%

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

       5. associate-/r* 73.9%

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

   13. Simplified 73.9%

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

   if 1.34999999999999997e97 < t

   1. Initial program 41.9%

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

   2. Taylor expanded in t around 0 48.3%

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

      1. times-frac 47.1%

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

      2. unpow2 47.1%

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

      3. unpow2 47.1%

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

      4. times-frac 56.0%

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

   4. Simplified 56.0%

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

      1. associate-*l/ 56.0%

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

   6. Applied egg-rr 56.0%

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

   7. Taylor expanded in k around 0 59.5%

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

      1. associate-/l* 56.3%

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

      2. unpow2 56.3%

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

   9. Simplified 56.3%

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

       1. frac-times 59.5%

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

       2. div-inv 59.5%

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

       3. clear-num 59.5%

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

       4. *-commutative 59.5%

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

   11. Applied egg-rr 59.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{t}, 0.3333333333333333, \frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}{\ell \cdot \cos k}}\\ \end{array} \]

Alternative 12: 69.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot \left(k \cdot \frac{t}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.5e-62)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (if (<= t 1.05e-74)
     (/ 2.0 (* (/ (* k k) (cos k)) (/ (* k (* k (/ t l))) l)))
     (if (<= t 1.35e+97)
       (* (/ l k) (/ (/ l k) (pow t 3.0)))
       (/ 2.0 (/ (* (* k k) (* (/ t l) (* k k))) (* l (cos k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.5e-62) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else if (t <= 1.05e-74) {
		tmp = 2.0 / (((k * k) / cos(k)) * ((k * (k * (t / l))) / l));
	} else if (t <= 1.35e+97) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k * k) * ((t / l) * (k * k))) / (l * cos(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 <= (-1.5d-62)) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else if (t <= 1.05d-74) then
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((k * (k * (t / l))) / l))
    else if (t <= 1.35d+97) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((k * k) * ((t / l) * (k * k))) / (l * cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.5e-62) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else if (t <= 1.05e-74) {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((k * (k * (t / l))) / l));
	} else if (t <= 1.35e+97) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k * k) * ((t / l) * (k * k))) / (l * Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.5e-62:
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	elif t <= 1.05e-74:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((k * (k * (t / l))) / l))
	elif t <= 1.35e+97:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (((k * k) * ((t / l) * (k * k))) / (l * math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.5e-62)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	elseif (t <= 1.05e-74)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(k * Float64(k * Float64(t / l))) / l)));
	elseif (t <= 1.35e+97)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k * k))) / Float64(l * cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.5e-62)
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	elseif (t <= 1.05e-74)
		tmp = 2.0 / (((k * k) / cos(k)) * ((k * (k * (t / l))) / l));
	elseif (t <= 1.35e+97)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((k * k) * ((t / l) * (k * k))) / (l * cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.5e-62], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-74], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+97], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.5000000000000001e-62

   1. Initial program 66.8%

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

      1. associate-/r* 66.8%

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

      2. associate-*l* 55.4%

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

      3. sqr-neg 55.4%

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

      4. associate-*l* 66.8%

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

      5. *-commutative 66.8%

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

      6. sqr-neg 66.8%

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

      7. associate-*l/ 66.8%

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

      8. associate-*r/ 64.9%

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

      9. associate-/r/ 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in k around 0 51.6%

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

      1. unpow2 51.6%

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

      2. *-commutative 51.6%

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

      3. unpow2 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in l around 0 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. associate-*l* 62.0%

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

      4. times-frac 68.7%

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

   9. Simplified 68.7%

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

   if -1.5000000000000001e-62 < t < 1.05e-74

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0 61.4%

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

      1. times-frac 62.8%

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

      2. unpow2 62.8%

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

      3. unpow2 62.8%

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

      4. times-frac 83.6%

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

   4. Simplified 83.6%

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

      1. associate-*l/ 83.5%

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

   6. Applied egg-rr 83.5%

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

   7. Taylor expanded in k around 0 62.2%

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

      1. associate-/l* 69.0%

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

      2. unpow2 69.0%

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

   9. Simplified 69.0%

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

   10. Taylor expanded in k around 0 62.2%

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

       1. associate-*r/ 69.0%

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

       2. unpow2 69.0%

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

       3. associate-*l* 69.0%

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

   12. Simplified 69.0%

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

   if 1.05e-74 < t < 1.34999999999999997e97

   1. Initial program 54.6%

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

      1. associate-/r* 53.0%

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

      2. associate-*l* 46.0%

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

      3. sqr-neg 46.0%

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

      4. associate-*l* 53.0%

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

      5. *-commutative 53.0%

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

      6. sqr-neg 53.0%

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

      7. associate-*l/ 62.5%

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

      8. associate-*r/ 57.2%

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

      9. associate-/r/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in k around 0 38.0%

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

      1. unpow2 38.0%

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

      2. *-commutative 38.0%

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

      3. unpow2 38.0%

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

   6. Simplified 38.0%

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

      1. add-cbrt-cube 34.8%

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

      2. times-frac 34.6%

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

      3. times-frac 34.6%

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

      4. times-frac 38.6%

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

   8. Applied egg-rr 38.6%

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

      1. associate-*l* 38.6%

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

      2. cube-unmult 38.6%

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

      3. *-commutative 38.6%

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

      4. unpow2 38.6%

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

      5. times-frac 34.8%

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

      6. unpow2 34.8%

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

      7. associate-*l* 44.6%

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

      8. times-frac 48.5%

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

   10. Simplified 48.5%

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

   11. Taylor expanded in l around 0 38.0%

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

       1. unpow2 38.0%

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

       2. unpow2 38.0%

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

       3. associate-*r* 50.9%

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

       4. times-frac 67.7%

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

       5. associate-/r* 73.9%

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

   13. Simplified 73.9%

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

   if 1.34999999999999997e97 < t

   1. Initial program 41.9%

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

   2. Taylor expanded in t around 0 48.3%

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

      1. times-frac 47.1%

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

      2. unpow2 47.1%

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

      3. unpow2 47.1%

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

      4. times-frac 56.0%

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

   4. Simplified 56.0%

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

      1. associate-*l/ 56.0%

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

   6. Applied egg-rr 56.0%

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

   7. Taylor expanded in k around 0 59.5%

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

      1. associate-/l* 56.3%

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

      2. unpow2 56.3%

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

   9. Simplified 56.3%

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

       1. frac-times 59.5%

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

       2. div-inv 59.5%

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

       3. clear-num 59.5%

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

       4. *-commutative 59.5%

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

   11. Applied egg-rr 59.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot \left(k \cdot \frac{t}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}{\ell \cdot \cos k}}\\ \end{array} \]

Alternative 13: 65.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.4e-83)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l t) (/ l (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.4e-83) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.4d-83) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.4e-83) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.4e-83:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.4e-83)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.4e-83)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.4e-83], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3999999999999998e-83

   1. Initial program 57.5%

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

      1. associate-/r* 57.2%

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

      2. associate-*l* 49.1%

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

      3. sqr-neg 49.1%

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

      4. associate-*l* 57.2%

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

      5. *-commutative 57.2%

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

      6. sqr-neg 57.2%

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

      7. associate-*l/ 58.8%

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

      8. associate-*r/ 56.9%

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

      9. associate-/r/ 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. unpow2 48.4%

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

      2. *-commutative 48.4%

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

      3. unpow2 48.4%

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

   6. Simplified 48.4%

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

      1. add-cbrt-cube 47.9%

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

      2. times-frac 47.8%

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

      3. times-frac 47.8%

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

      4. times-frac 52.1%

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

   8. Applied egg-rr 52.1%

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

      1. associate-*l* 52.1%

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

      2. cube-unmult 52.1%

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

      3. *-commutative 52.1%

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

      4. unpow2 52.1%

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

      5. times-frac 47.9%

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

      6. unpow2 47.9%

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

      7. associate-*l* 56.0%

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

      8. times-frac 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in l around 0 48.4%

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

       1. unpow2 48.4%

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

       2. unpow2 48.4%

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

       3. associate-*r* 57.5%

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

       4. times-frac 65.6%

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

       5. associate-/r* 67.1%

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

   13. Simplified 67.1%

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

   if 3.3999999999999998e-83 < k

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. associate-*l* 35.0%

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

      3. sqr-neg 35.0%

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

      4. associate-*l* 35.0%

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

      5. *-commutative 35.0%

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

      6. sqr-neg 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-*r/ 34.9%

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

      9. associate-/r/ 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in k around inf 49.3%

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

      1. times-frac 47.5%

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

      2. unpow2 47.5%

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

      3. unpow2 47.5%

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

      4. *-commutative 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in k around 0 39.4%

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

      1. unpow2 39.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

      2. times-frac 51.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{t}\right)}\right) \]

      3. unpow2 51.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t}\right)\right) \]

   9. Simplified 51.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 14: 65.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-83)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (/ (* k k) (cos k)) (* (/ t l) (/ (* k k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-83) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * ((k * k) / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.7d-83) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((t / l) * ((k * k) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-83) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((t / l) * ((k * k) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-83:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((t / l) * ((k * k) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-83)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(t / l) * Float64(Float64(k * k) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-83)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * ((k * k) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-83], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999991e-83

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 57.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 57.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 57.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 57.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 58.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 56.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 56.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 56.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 48.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 48.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 48.4%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 48.4%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 47.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}}} \]

      2. times-frac 47.8%

         \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      3. times-frac 47.8%

         \[\leadsto \sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      4. times-frac 52.1%

         \[\leadsto \sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}} \]

   8. Applied egg-rr 52.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 52.1%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right)}} \]

      2. cube-unmult 52.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}^{3}}} \]

      3. *-commutative 52.1%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\right)}}^{3}} \]

      4. unpow2 52.1%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell}{\color{blue}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}}\right)}^{3}} \]

      5. times-frac 47.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{3}}\right)}}^{3}} \]

      6. unpow2 47.9%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{3}} \]

      7. associate-*l* 56.0%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{3}} \]

      8. times-frac 60.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}}^{3}} \]

   10. Simplified 60.3%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}^{3}}} \]

   11. Taylor expanded in l around 0 48.4%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

       1. unpow2 48.4%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 48.4%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.5%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.6%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 67.1%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   13. Simplified 67.1%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 2.69999999999999991e-83 < k

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 49.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. times-frac 47.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 47.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]

      3. unpow2 47.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      4. times-frac 62.5%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

   4. Simplified 62.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

   5. Taylor expanded in k around 0 39.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 39.4%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      2. times-frac 51.0%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

      3. unpow2 51.0%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   7. Simplified 51.0%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 15: 63.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.4e-68)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.4d-68) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e-68) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.4e-68:
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.4e-68)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.4e-68)
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.4e-68], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999991e-68

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 57.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 57.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 57.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 57.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 58.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 56.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 56.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 49.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 49.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   7. Taylor expanded in l around 0 49.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 49.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*l* 58.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 66.0%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if 2.39999999999999991e-68 < k

   1. Initial program 34.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 34.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 34.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 34.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 34.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 34.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 34.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 34.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 34.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 34.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 34.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 48.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 46.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 46.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 46.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 46.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 46.1%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 34.2%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 34.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 34.2%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 48.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 48.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 16: 64.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e-68)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-68) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-68) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-68) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.2e-68:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e-68)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e-68)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.2e-68], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000002e-68

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 57.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 57.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 57.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 57.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 58.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 56.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 56.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 49.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 49.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 48.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}}} \]

      2. times-frac 48.3%

         \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      3. times-frac 48.3%

         \[\leadsto \sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      4. times-frac 52.6%

         \[\leadsto \sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}} \]

   8. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 52.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right)}} \]

      2. cube-unmult 52.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}^{3}}} \]

      3. *-commutative 52.6%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\right)}}^{3}} \]

      4. unpow2 52.6%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell}{\color{blue}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}}\right)}^{3}} \]

      5. times-frac 48.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{3}}\right)}}^{3}} \]

      6. unpow2 48.5%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{3}} \]

      7. associate-*l* 56.4%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{3}} \]

      8. times-frac 60.7%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}}^{3}} \]

   10. Simplified 60.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}^{3}}} \]

   11. Taylor expanded in l around 0 49.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

       1. unpow2 49.0%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 49.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 58.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 66.0%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 67.5%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   13. Simplified 67.5%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 2.20000000000000002e-68 < k

   1. Initial program 34.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 34.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 34.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 34.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 34.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 34.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 34.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 34.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 34.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 34.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 34.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 48.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. times-frac 46.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

      2. unpow2 46.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

      3. unpow2 46.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 46.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 46.1%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 34.2%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 34.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 34.2%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 48.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 48.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 17: 64.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.85e-68)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-68) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.85d-68) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-68) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.85e-68:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.85e-68)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.85e-68)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.85e-68], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85000000000000001e-68

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 57.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 57.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 57.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 57.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 58.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 56.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 56.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 49.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 49.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 49.0%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 48.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}}} \]

      2. times-frac 48.3%

         \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      3. times-frac 48.3%

         \[\leadsto \sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}\right) \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

      4. times-frac 52.6%

         \[\leadsto \sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}} \]

   8. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 52.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)\right)}} \]

      2. cube-unmult 52.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\right)}^{3}}} \]

      3. *-commutative 52.6%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\right)}}^{3}} \]

      4. unpow2 52.6%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell}{\color{blue}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}}\right)}^{3}} \]

      5. times-frac 48.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{3}}\right)}}^{3}} \]

      6. unpow2 48.5%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}\right)}^{3}} \]

      7. associate-*l* 56.4%

         \[\leadsto \sqrt[3]{{\left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}\right)}^{3}} \]

      8. times-frac 60.7%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}}^{3}} \]

   10. Simplified 60.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\right)}^{3}}} \]

   11. Taylor expanded in l around 0 49.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   12. Step-by-step derivation

       1. unpow2 49.0%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 49.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 58.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 66.0%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 67.5%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   13. Simplified 67.5%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 1.85000000000000001e-68 < k

   1. Initial program 34.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 48.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
   3. Step-by-step derivation

      1. times-frac 46.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 46.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]

      3. unpow2 46.5%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      4. times-frac 61.6%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

   4. Simplified 61.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

   5. Taylor expanded in k around 0 34.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 34.2%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      2. times-frac 48.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

   7. Simplified 48.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 18: 55.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* l (/ l (* t (pow k 4.0))))))

C

double code(double t, double l, double k) {
	return 2.0 * (l * (l / (t * pow(k, 4.0))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l * (l / (t * (k ** 4.0d0))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (l * (l / (t * Math.pow(k, 4.0))));
}

Python

def code(t, l, k):
	return 2.0 * (l * (l / (t * math.pow(k, 4.0))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(l * Float64(l / Float64(t * (k ^ 4.0)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (l * (l / (t * (k ^ 4.0))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(l * N[(l / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 51.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}} \]
3. Step-by-step derivation

   1. times-frac 52.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]

   2. unpow2 52.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]

   3. unpow2 52.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   4. times-frac 62.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

4. Simplified 62.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}} \]

5. Taylor expanded in k around 0 43.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. unpow2 43.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

7. Simplified 43.9%

   \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot t}} \]

8. Taylor expanded in l around 0 43.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
9. Step-by-step derivation

   1. *-rgt-identity 43.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{4} \cdot t} \]

   2. *-commutative 43.9%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot 1}{\color{blue}{t \cdot {k}^{4}}} \]

   3. associate-*r/ 43.7%

      \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)} \]

   4. unpow2 43.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{t \cdot {k}^{4}}\right) \]

   5. associate-*l* 49.1%

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{1}{t \cdot {k}^{4}}\right)\right)} \]

   6. associate-*r/ 49.3%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot 1}{t \cdot {k}^{4}}}\right) \]

   7. *-rgt-identity 49.3%

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\color{blue}{\ell}}{t \cdot {k}^{4}}\right) \]

   8. *-commutative 49.3%

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}}\right) \]

10. Simplified 49.3%

    \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]

11. Final simplification 49.3%

    \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \]

Alternative 19: 56.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 50.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 44.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 44.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 50.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 50.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 50.4%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 51.6%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 50.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 50.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 50.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 51.1%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. times-frac 51.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]

   2. unpow2 51.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]

   3. unpow2 51.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]

   4. *-commutative 51.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

6. Simplified 51.7%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 43.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 43.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 43.9%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 51.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 51.3%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Final simplification 51.3%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Reproduce

herbie shell --seed 2023264 
(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))))