Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 88.2%

Time: 30.3s

Alternatives: 21

Speedup: 3.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} t_1 := \sqrt{\sqrt[3]{\ell}}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{t_1 \cdot t_1}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{t_2}, \left(\ell \cdot \frac{\ell}{{k}^{4}}\right) \cdot \left(\frac{t \cdot \cos k}{t_2} \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sqrt (cbrt l))) (t_2 (pow (sin k) 2.0)))
   (if (<= k 7.5e-16)
     (/
      (pow
       (/
        (cbrt (/ 2.0 (tan k)))
        (/ t (/ (* t_1 t_1) (/ (cbrt (sin k)) (cbrt l)))))
       3.0)
      (+ 2.0 (pow (/ k t) 2.0)))
     (if (<= k 2.35e+154)
       (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) t_2)))
       (fma
        2.0
        (* (* (/ l k) (/ l k)) (/ (/ (cos k) t) t_2))
        (* (* l (/ l (pow k 4.0))) (* (/ (* t (cos k)) t_2) -4.0)))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double t_1 = sqrt(cbrt(l));
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 7.5e-16) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / ((t_1 * t_1) / (cbrt(sin(k)) / cbrt(l))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else if (k <= 2.35e+154) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / t_2));
	} else {
		tmp = fma(2.0, (((l / k) * (l / k)) * ((cos(k) / t) / t_2)), ((l * (l / pow(k, 4.0))) * (((t * cos(k)) / t_2) * -4.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	t_1 = sqrt(cbrt(l))
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 7.5e-16)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(Float64(t_1 * t_1) / Float64(cbrt(sin(k)) / cbrt(l))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (k <= 2.35e+154)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / t_2)));
	else
		tmp = fma(2.0, Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(cos(k) / t) / t_2)), Float64(Float64(l * Float64(l / (k ^ 4.0))) * Float64(Float64(Float64(t * cos(k)) / t_2) * -4.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Sqrt[N[Power[l, 1/3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 7.5e-16], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.35e+154], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.5e-16

   1. Initial program 58.5%

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

      1. associate-/r* 58.5%

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

      2. associate-*l* 53.5%

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

      3. sqr-neg 53.5%

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

      4. associate-*l* 58.5%

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

      5. *-commutative 58.5%

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

      6. sqr-neg 58.5%

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

      7. associate-/r* 58.5%

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

   3. Simplified 56.8%

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

      1. associate-/l/ 56.8%

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

      2. associate-/r/ 58.5%

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

      3. add-cube-cbrt 58.4%

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

      4. pow3 58.4%

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

   5. Applied egg-rr 69.6%

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

      1. cbrt-div 81.2%

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

   7. Applied egg-rr 81.2%

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

      1. cbrt-div 85.2%

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

   9. Applied egg-rr 85.2%

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

       1. add-sqr-sqrt 40.7%

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

   11. Applied egg-rr 40.7%

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

   if 7.5e-16 < k < 2.34999999999999992e154

   1. Initial program 39.5%

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

      1. associate-/r* 39.6%

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

      2. associate-*l* 39.6%

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

      3. sqr-neg 39.6%

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

      4. associate-*l* 39.6%

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

      5. *-commutative 39.6%

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

      6. sqr-neg 39.6%

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

      7. associate-*l/ 39.5%

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

      8. associate-*r/ 39.6%

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

      9. associate-/r/ 39.4%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in k around inf 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-*r* 78.7%

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

      3. times-frac 78.7%

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

      4. unpow2 78.7%

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

      5. associate-*l* 78.7%

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

      6. unpow2 78.7%

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

   6. Simplified 78.7%

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

      1. frac-times 78.7%

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

   8. Applied egg-rr 78.7%

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

   9. Taylor expanded in k around inf 78.7%

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

       1. associate-*r* 78.7%

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

       2. unpow2 78.7%

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

       3. associate-*r* 78.7%

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

       4. times-frac 78.6%

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

       5. associate-*r* 78.6%

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

       6. unpow2 78.6%

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

       7. unpow2 78.6%

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

       8. *-commutative 78.6%

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

       9. times-frac 82.2%

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

       10. unpow2 82.2%

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

   11. Simplified 82.2%

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

   if 2.34999999999999992e154 < k

   1. Initial program 45.4%

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

      1. associate-/r* 45.4%

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

      2. associate-*l* 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-*l* 45.4%

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

      5. *-commutative 45.4%

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

      6. sqr-neg 45.4%

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

      7. associate-/r* 45.4%

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

   3. Simplified 45.4%

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

      1. associate-/l/ 45.4%

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

      2. associate-/r/ 45.4%

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

      3. add-cube-cbrt 45.4%

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

      4. pow3 45.4%

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

   5. Applied egg-rr 60.3%

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

      1. cbrt-div 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in k around inf 62.2%

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

      1. +-commutative 62.2%

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

      2. fma-def 62.2%

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

      3. times-frac 62.2%

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

      4. unpow2 62.2%

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

      5. unpow2 62.2%

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

      6. times-frac 69.3%

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

      7. associate-/r* 69.3%

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

      8. *-commutative 69.3%

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

      9. times-frac 76.8%

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

      10. unpow2 76.8%

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

      11. associate-*r/ 96.7%

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

   10. Simplified 96.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}, \left(\ell \cdot \frac{\ell}{{k}^{4}}\right) \cdot \left(\frac{t \cdot \cos k}{{\sin k}^{2}} \cdot -4\right)\right)\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.5× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_1 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        INFINITY)
     (/ (* (/ (* 2.0 l) (* (tan k) (pow t 3.0))) (/ l (sin k))) (+ 2.0 t_1))
     (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.5%

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

      1. associate-/r* 83.5%

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

      2. associate-*l* 78.0%

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

      3. sqr-neg 78.0%

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

      4. associate-*l* 83.5%

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

      5. *-commutative 83.5%

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

      6. sqr-neg 83.5%

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

      7. associate-*l/ 84.0%

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

      8. associate-*r/ 82.8%

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

      9. associate-/r/ 82.8%

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

   3. Simplified 82.8%

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

      1. expm1-log1p-u 60.1%

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

      2. expm1-udef 55.6%

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

      3. associate-*l/ 55.6%

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

      4. associate-*r* 55.6%

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

   5. Applied egg-rr 55.6%

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

      1. expm1-def 60.1%

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

      2. expm1-log1p 82.8%

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

      3. associate-*r* 82.8%

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

      4. times-frac 87.3%

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

      5. *-commutative 87.3%

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

   7. Simplified 87.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. associate-*l* 0.0%

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

      3. sqr-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. *-commutative 0.0%

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

      6. sqr-neg 0.0%

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

      7. associate-*l/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. associate-/r/ 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-*r* 47.1%

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

      3. times-frac 47.1%

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

      4. unpow2 47.1%

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

      5. associate-*l* 51.4%

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

      6. unpow2 51.4%

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

   6. Simplified 51.4%

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

      1. frac-times 51.4%

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

   8. Applied egg-rr 51.4%

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

   9. Taylor expanded in k around inf 47.1%

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

       1. associate-*r* 47.1%

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

       2. unpow2 47.1%

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

       3. associate-*r* 51.4%

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

       4. times-frac 51.4%

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

       5. associate-*r* 47.1%

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

       6. unpow2 47.1%

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

       7. unpow2 47.1%

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

       8. *-commutative 47.1%

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

       9. times-frac 62.8%

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

       10. unpow2 62.8%

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

   11. Simplified 62.8%

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

4. Final simplification 78.7%

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

Alternative 3: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 3 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{t_1}, \left(\ell \cdot \frac{\ell}{{k}^{4}}\right) \cdot \left(\frac{t \cdot \cos k}{t_1} \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 3e-16)
     (/
      (pow
       (/
        (cbrt (/ 2.0 (tan k)))
        (/ t (/ (cbrt l) (/ (cbrt (sin k)) (cbrt l)))))
       3.0)
      (+ 2.0 (pow (/ k t) 2.0)))
     (if (<= k 2.35e+154)
       (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) t_1)))
       (fma
        2.0
        (* (* (/ l k) (/ l k)) (/ (/ (cos k) t) t_1))
        (* (* l (/ l (pow k 4.0))) (* (/ (* t (cos k)) t_1) -4.0)))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 3e-16) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / (cbrt(sin(k)) / cbrt(l))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else if (k <= 2.35e+154) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / t_1));
	} else {
		tmp = fma(2.0, (((l / k) * (l / k)) * ((cos(k) / t) / t_1)), ((l * (l / pow(k, 4.0))) * (((t * cos(k)) / t_1) * -4.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 3e-16)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / Float64(cbrt(sin(k)) / cbrt(l))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (k <= 2.35e+154)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / t_1)));
	else
		tmp = fma(2.0, Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(cos(k) / t) / t_1)), Float64(Float64(l * Float64(l / (k ^ 4.0))) * Float64(Float64(Float64(t * cos(k)) / t_1) * -4.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 3e-16], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.35e+154], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.99999999999999994e-16

   1. Initial program 58.5%

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

      1. associate-/r* 58.5%

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

      2. associate-*l* 53.5%

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

      3. sqr-neg 53.5%

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

      4. associate-*l* 58.5%

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

      5. *-commutative 58.5%

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

      6. sqr-neg 58.5%

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

      7. associate-/r* 58.5%

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

   3. Simplified 56.8%

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

      1. associate-/l/ 56.8%

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

      2. associate-/r/ 58.5%

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

      3. add-cube-cbrt 58.4%

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

      4. pow3 58.4%

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

   5. Applied egg-rr 69.6%

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

      1. cbrt-div 81.2%

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

   7. Applied egg-rr 81.2%

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

      1. cbrt-div 85.2%

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

   9. Applied egg-rr 85.2%

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

   if 2.99999999999999994e-16 < k < 2.34999999999999992e154

   1. Initial program 39.5%

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

      1. associate-/r* 39.6%

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

      2. associate-*l* 39.6%

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

      3. sqr-neg 39.6%

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

      4. associate-*l* 39.6%

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

      5. *-commutative 39.6%

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

      6. sqr-neg 39.6%

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

      7. associate-*l/ 39.5%

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

      8. associate-*r/ 39.6%

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

      9. associate-/r/ 39.4%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in k around inf 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-*r* 78.7%

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

      3. times-frac 78.7%

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

      4. unpow2 78.7%

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

      5. associate-*l* 78.7%

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

      6. unpow2 78.7%

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

   6. Simplified 78.7%

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

      1. frac-times 78.7%

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

   8. Applied egg-rr 78.7%

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

   9. Taylor expanded in k around inf 78.7%

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

       1. associate-*r* 78.7%

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

       2. unpow2 78.7%

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

       3. associate-*r* 78.7%

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

       4. times-frac 78.6%

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

       5. associate-*r* 78.6%

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

       6. unpow2 78.6%

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

       7. unpow2 78.6%

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

       8. *-commutative 78.6%

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

       9. times-frac 82.2%

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

       10. unpow2 82.2%

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

   11. Simplified 82.2%

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

   if 2.34999999999999992e154 < k

   1. Initial program 45.4%

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

      1. associate-/r* 45.4%

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

      2. associate-*l* 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-*l* 45.4%

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

      5. *-commutative 45.4%

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

      6. sqr-neg 45.4%

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

      7. associate-/r* 45.4%

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

   3. Simplified 45.4%

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

      1. associate-/l/ 45.4%

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

      2. associate-/r/ 45.4%

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

      3. add-cube-cbrt 45.4%

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

      4. pow3 45.4%

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

   5. Applied egg-rr 60.3%

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

      1. cbrt-div 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in k around inf 62.2%

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

      1. +-commutative 62.2%

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

      2. fma-def 62.2%

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

      3. times-frac 62.2%

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

      4. unpow2 62.2%

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

      5. unpow2 62.2%

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

      6. times-frac 69.3%

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

      7. associate-/r* 69.3%

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

      8. *-commutative 69.3%

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

      9. times-frac 76.8%

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

      10. unpow2 76.8%

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

      11. associate-*r/ 96.7%

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

   10. Simplified 96.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}, \left(\ell \cdot \frac{\ell}{{k}^{4}}\right) \cdot \left(\frac{t \cdot \cos k}{{\sin k}^{2}} \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-31} \lor \neg \left(t \leq 1.6 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1e-31) (not (<= t 1.6e-34)))
   (/
    (pow
     (/ (cbrt (/ 2.0 (tan k))) (/ t (/ (cbrt l) (/ (cbrt (sin k)) (cbrt l)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1e-31) || !(t <= 1.6e-34)) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / (cbrt(sin(k)) / cbrt(l))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1e-31) || !(t <= 1.6e-34)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / (Math.cbrt(l) / (Math.cbrt(Math.sin(k)) / Math.cbrt(l))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1e-31) || !(t <= 1.6e-34))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / Float64(cbrt(sin(k)) / cbrt(l))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1e-31], N[Not[LessEqual[t, 1.6e-34]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1e-31 or 1.60000000000000001e-34 < t

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

      2. associate-*l* 59.4%

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

      3. sqr-neg 59.4%

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

      4. associate-*l* 66.1%

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

      5. *-commutative 66.1%

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

      6. sqr-neg 66.1%

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

      7. associate-/r* 66.1%

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

   3. Simplified 63.8%

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

      1. associate-/l/ 63.7%

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

      2. associate-/r/ 66.1%

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

      3. add-cube-cbrt 65.9%

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

      4. pow3 65.9%

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

   5. Applied egg-rr 71.5%

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

      1. cbrt-div 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. cbrt-div 95.8%

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

   9. Applied egg-rr 95.8%

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

   if -1e-31 < t < 1.60000000000000001e-34

   1. Initial program 40.6%

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

      1. associate-/r* 40.6%

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

      2. associate-*l* 40.6%

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

      3. sqr-neg 40.6%

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

      4. associate-*l* 40.6%

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

      5. *-commutative 40.6%

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

      6. sqr-neg 40.6%

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

      7. associate-*l/ 40.6%

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

      8. associate-*r/ 40.5%

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

      9. associate-/r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in k around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r* 75.3%

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

      3. times-frac 75.4%

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

      4. unpow2 75.4%

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

      5. associate-*l* 78.6%

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

      6. unpow2 78.6%

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

   6. Simplified 78.6%

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

      1. frac-times 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in k around inf 75.4%

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

       1. associate-*r* 75.3%

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

       2. unpow2 75.3%

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

       3. associate-*r* 78.6%

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

       4. times-frac 78.6%

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

       5. associate-*r* 75.4%

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

       6. unpow2 75.4%

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

       7. unpow2 75.4%

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

       8. *-commutative 75.4%

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

       9. times-frac 85.8%

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

       10. unpow2 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-31} \lor \neg \left(t \leq 1.6 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-32} \lor \neg \left(t \leq 7.4 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{1}{\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{\sin k}{\ell}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -8.2e-32) (not (<= t 7.4e-37)))
   (/
    (pow
     (* (cbrt (/ 2.0 (tan k))) (/ 1.0 (* (/ t (cbrt l)) (cbrt (/ (sin k) l)))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.2e-32) || !(t <= 7.4e-37)) {
		tmp = pow((cbrt((2.0 / tan(k))) * (1.0 / ((t / cbrt(l)) * cbrt((sin(k) / l))))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.2e-32) || !(t <= 7.4e-37)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) * (1.0 / ((t / Math.cbrt(l)) * Math.cbrt((Math.sin(k) / l))))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -8.2e-32) || !(t <= 7.4e-37))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) * Float64(1.0 / Float64(Float64(t / cbrt(l)) * cbrt(Float64(sin(k) / l))))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -8.2e-32], N[Not[LessEqual[t, 7.4e-37]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.1999999999999995e-32 or 7.4e-37 < t

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

      2. associate-*l* 59.4%

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

      3. sqr-neg 59.4%

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

      4. associate-*l* 66.1%

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

      5. *-commutative 66.1%

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

      6. sqr-neg 66.1%

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

      7. associate-/r* 66.1%

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

   3. Simplified 63.8%

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

      1. associate-/l/ 63.7%

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

      2. associate-/r/ 66.1%

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

      3. add-cube-cbrt 65.9%

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

      4. pow3 65.9%

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

   5. Applied egg-rr 71.5%

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

      1. cbrt-div 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. cbrt-div 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. div-inv 95.9%

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

       2. associate-/r/ 95.9%

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

       3. cbrt-undiv 90.5%

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

   11. Applied egg-rr 90.5%

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

   if -8.1999999999999995e-32 < t < 7.4e-37

   1. Initial program 40.6%

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

      1. associate-/r* 40.6%

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

      2. associate-*l* 40.6%

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

      3. sqr-neg 40.6%

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

      4. associate-*l* 40.6%

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

      5. *-commutative 40.6%

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

      6. sqr-neg 40.6%

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

      7. associate-*l/ 40.6%

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

      8. associate-*r/ 40.5%

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

      9. associate-/r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in k around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r* 75.3%

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

      3. times-frac 75.4%

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

      4. unpow2 75.4%

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

      5. associate-*l* 78.6%

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

      6. unpow2 78.6%

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

   6. Simplified 78.6%

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

      1. frac-times 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in k around inf 75.4%

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

       1. associate-*r* 75.3%

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

       2. unpow2 75.3%

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

       3. associate-*r* 78.6%

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

       4. times-frac 78.6%

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

       5. associate-*r* 75.4%

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

       6. unpow2 75.4%

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

       7. unpow2 75.4%

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

       8. *-commutative 75.4%

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

       9. times-frac 85.8%

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

       10. unpow2 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-32} \lor \neg \left(t \leq 7.4 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{1}{\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{\sin k}{\ell}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-31} \lor \neg \left(t \leq 1.7 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.2e-31) (not (<= t 1.7e-33)))
   (/
    (pow
     (* (/ (cbrt (/ 2.0 (tan k))) t) (/ (cbrt l) (cbrt (/ (sin k) l))))
     3.0)
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.2e-31) || !(t <= 1.7e-33)) {
		tmp = pow(((cbrt((2.0 / tan(k))) / t) * (cbrt(l) / cbrt((sin(k) / l)))), 3.0) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.2e-31) || !(t <= 1.7e-33)) {
		tmp = Math.pow(((Math.cbrt((2.0 / Math.tan(k))) / t) * (Math.cbrt(l) / Math.cbrt((Math.sin(k) / l)))), 3.0) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.2e-31) || !(t <= 1.7e-33))
		tmp = Float64((Float64(Float64(cbrt(Float64(2.0 / tan(k))) / t) * Float64(cbrt(l) / cbrt(Float64(sin(k) / l)))) ^ 3.0) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.2e-31], N[Not[LessEqual[t, 1.7e-33]], $MachinePrecision]], N[(N[Power[N[(N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2e-31 or 1.7e-33 < t

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

      2. associate-*l* 59.4%

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

      3. sqr-neg 59.4%

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

      4. associate-*l* 66.1%

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

      5. *-commutative 66.1%

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

      6. sqr-neg 66.1%

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

      7. associate-/r* 66.1%

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

   3. Simplified 63.8%

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

      1. associate-/l/ 63.7%

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

      2. associate-/r/ 66.1%

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

      3. add-cube-cbrt 65.9%

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

      4. pow3 65.9%

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

   5. Applied egg-rr 71.5%

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

      1. cbrt-div 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. cbrt-div 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. div-inv 95.8%

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

       2. associate-/r/ 95.8%

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

       3. cbrt-undiv 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. associate-*r/ 90.4%

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

       2. associate-*l/ 90.4%

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

       3. *-rgt-identity 90.4%

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

   13. Simplified 90.4%

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

   if -1.2e-31 < t < 1.7e-33

   1. Initial program 40.6%

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

      1. associate-/r* 40.6%

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

      2. associate-*l* 40.6%

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

      3. sqr-neg 40.6%

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

      4. associate-*l* 40.6%

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

      5. *-commutative 40.6%

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

      6. sqr-neg 40.6%

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

      7. associate-*l/ 40.6%

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

      8. associate-*r/ 40.5%

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

      9. associate-/r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in k around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r* 75.3%

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

      3. times-frac 75.4%

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

      4. unpow2 75.4%

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

      5. associate-*l* 78.6%

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

      6. unpow2 78.6%

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

   6. Simplified 78.6%

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

      1. frac-times 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in k around inf 75.4%

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

       1. associate-*r* 75.3%

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

       2. unpow2 75.3%

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

       3. associate-*r* 78.6%

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

       4. times-frac 78.6%

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

       5. associate-*r* 75.4%

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

       6. unpow2 75.4%

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

       7. unpow2 75.4%

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

       8. *-commutative 75.4%

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

       9. times-frac 85.8%

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

       10. unpow2 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-31} \lor \neg \left(t \leq 1.7 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{t} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 7: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-32} \lor \neg \left(t \leq 7.4 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -8.2e-32) (not (<= t 7.4e-37)))
   (/
    (pow
     (/ (cbrt (/ 2.0 (tan k))) (/ t (/ (cbrt l) (cbrt (/ (sin k) l)))))
     3.0)
    (+ 2.0 (* (/ k t) (/ k t))))
   (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.2e-32) || !(t <= 7.4e-37)) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / (cbrt(l) / cbrt((sin(k) / l))))), 3.0) / (2.0 + ((k / t) * (k / t)));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.2e-32) || !(t <= 7.4e-37)) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / (Math.cbrt(l) / Math.cbrt((Math.sin(k) / l))))), 3.0) / (2.0 + ((k / t) * (k / t)));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -8.2e-32) || !(t <= 7.4e-37))
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / Float64(cbrt(l) / cbrt(Float64(sin(k) / l))))) ^ 3.0) / Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -8.2e-32], N[Not[LessEqual[t, 7.4e-37]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.1999999999999995e-32 or 7.4e-37 < t

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

      2. associate-*l* 59.4%

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

      3. sqr-neg 59.4%

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

      4. associate-*l* 66.1%

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

      5. *-commutative 66.1%

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

      6. sqr-neg 66.1%

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

      7. associate-/r* 66.1%

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

   3. Simplified 63.8%

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

      1. associate-/l/ 63.7%

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

      2. associate-/r/ 66.1%

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

      3. add-cube-cbrt 65.9%

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

      4. pow3 65.9%

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

   5. Applied egg-rr 71.5%

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

      1. cbrt-div 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. unpow2 60.1%

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

   9. Applied egg-rr 90.4%

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

   if -8.1999999999999995e-32 < t < 7.4e-37

   1. Initial program 40.6%

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

      1. associate-/r* 40.6%

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

      2. associate-*l* 40.6%

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

      3. sqr-neg 40.6%

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

      4. associate-*l* 40.6%

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

      5. *-commutative 40.6%

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

      6. sqr-neg 40.6%

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

      7. associate-*l/ 40.6%

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

      8. associate-*r/ 40.5%

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

      9. associate-/r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in k around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r* 75.3%

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

      3. times-frac 75.4%

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

      4. unpow2 75.4%

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

      5. associate-*l* 78.6%

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

      6. unpow2 78.6%

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

   6. Simplified 78.6%

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

      1. frac-times 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in k around inf 75.4%

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

       1. associate-*r* 75.3%

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

       2. unpow2 75.3%

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

       3. associate-*r* 78.6%

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

       4. times-frac 78.6%

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

       5. associate-*r* 75.4%

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

       6. unpow2 75.4%

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

       7. unpow2 75.4%

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

       8. *-commutative 75.4%

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

       9. times-frac 85.8%

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

       10. unpow2 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-32} \lor \neg \left(t \leq 7.4 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 8: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-31} \lor \neg \left(t \leq 7.2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{3}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.85e-31) (not (<= t 7.2e-33)))
   (/
    (/ 2.0 (* (tan k) (* (sin k) (pow (/ t (* (cbrt l) (cbrt l))) 3.0))))
    (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
   (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.85e-31) || !(t <= 7.2e-33)) {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((t / (cbrt(l) * cbrt(l))), 3.0)))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.85e-31) || !(t <= 7.2e-33)) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((t / (Math.cbrt(l) * Math.cbrt(l))), 3.0)))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.85e-31) || !(t <= 7.2e-33))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64(t / Float64(cbrt(l) * cbrt(l))) ^ 3.0)))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.85e-31], N[Not[LessEqual[t, 7.2e-33]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8499999999999999e-31 or 7.20000000000000068e-33 < t

   1. Initial program 66.5%

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

      1. associate-/r* 66.6%

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

      2. +-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. add-cube-cbrt 66.4%

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

      2. pow3 66.4%

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

      3. cbrt-div 66.3%

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

      4. rem-cbrt-cube 71.8%

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

   5. Applied egg-rr 71.8%

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

      1. cbrt-prod 84.3%

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

   7. Applied egg-rr 84.3%

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

   if -1.8499999999999999e-31 < t < 7.20000000000000068e-33

   1. Initial program 40.3%

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

      1. associate-/r* 40.3%

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

      2. associate-*l* 40.3%

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

      3. sqr-neg 40.3%

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

      4. associate-*l* 40.3%

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

      5. *-commutative 40.3%

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

      6. sqr-neg 40.3%

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

      7. associate-*l/ 40.3%

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

      8. associate-*r/ 40.2%

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

      9. associate-/r/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in k around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-*r* 74.7%

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

      3. times-frac 74.8%

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

      4. unpow2 74.8%

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

      5. associate-*l* 78.0%

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

      6. unpow2 78.0%

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

   6. Simplified 78.0%

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

      1. frac-times 77.9%

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

   8. Applied egg-rr 77.9%

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

   9. Taylor expanded in k around inf 74.8%

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

       1. associate-*r* 74.7%

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

       2. unpow2 74.7%

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

       3. associate-*r* 77.9%

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

       4. times-frac 78.0%

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

       5. associate-*r* 74.8%

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

       6. unpow2 74.8%

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

       7. unpow2 74.8%

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

       8. *-commutative 74.8%

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

       9. times-frac 85.1%

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

       10. unpow2 85.1%

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

   11. Simplified 85.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-31} \lor \neg \left(t \leq 7.2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{3}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 9: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}{2 + t_1}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}{1 + \left(t_1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -8.8e-32)
     (/ (* (/ (* 2.0 l) (* (tan k) (pow t 3.0))) (/ l (sin k))) (+ 2.0 t_1))
     (if (<= t 1.9e-33)
       (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))
       (/
        (/ 2.0 (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))))
        (+ 1.0 (+ t_1 1.0)))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -8.8e-32) {
		tmp = (((2.0 * l) / (tan(k) * pow(t, 3.0))) * (l / sin(k))) / (2.0 + t_1);
	} else if (t <= 1.9e-33) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if (t <= (-8.8d-32)) then
        tmp = (((2.0d0 * l) / (tan(k) * (t ** 3.0d0))) * (l / sin(k))) / (2.0d0 + t_1)
    else if (t <= 1.9d-33) then
        tmp = 2.0d0 * (((l / t) * (l / (k * k))) * (cos(k) / (sin(k) ** 2.0d0)))
    else
        tmp = (2.0d0 / (tan(k) * (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0)))) / (1.0d0 + (t_1 + 1.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -8.8e-32) {
		tmp = (((2.0 * l) / (Math.tan(k) * Math.pow(t, 3.0))) * (l / Math.sin(k))) / (2.0 + t_1);
	} else if (t <= 1.9e-33) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if t <= -8.8e-32:
		tmp = (((2.0 * l) / (math.tan(k) * math.pow(t, 3.0))) * (l / math.sin(k))) / (2.0 + t_1)
	elif t <= 1.9e-33:
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (2.0 / (math.tan(k) * (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0)))) / (1.0 + (t_1 + 1.0))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -8.8e-32)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / Float64(tan(k) * (t ^ 3.0))) * Float64(l / sin(k))) / Float64(2.0 + t_1));
	elseif (t <= 1.9e-33)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))) / Float64(1.0 + Float64(t_1 + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if (t <= -8.8e-32)
		tmp = (((2.0 * l) / (tan(k) * (t ^ 3.0))) * (l / sin(k))) / (2.0 + t_1);
	elseif (t <= 1.9e-33)
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / (sin(k) ^ 2.0)));
	else
		tmp = (2.0 / (tan(k) * (sin(k) * (((t ^ 1.5) / l) ^ 2.0)))) / (1.0 + (t_1 + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -8.8e-32], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-33], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.7999999999999999e-32

   1. Initial program 69.9%

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

      1. associate-/r* 70.0%

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

      2. associate-*l* 61.2%

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

      3. sqr-neg 61.2%

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

      4. associate-*l* 70.0%

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

      5. *-commutative 70.0%

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

      6. sqr-neg 70.0%

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

      7. associate-*l/ 69.8%

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

      8. associate-*r/ 69.8%

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

      9. associate-/r/ 69.7%

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

   3. Simplified 69.7%

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

      1. expm1-log1p-u 57.4%

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

      2. expm1-udef 53.4%

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

      3. associate-*l/ 53.4%

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

      4. associate-*r* 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. expm1-def 57.3%

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

      2. expm1-log1p 69.7%

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

      3. associate-*r* 69.7%

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

      4. times-frac 77.4%

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

      5. *-commutative 77.4%

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

   7. Simplified 77.4%

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

   if -8.7999999999999999e-32 < t < 1.89999999999999997e-33

   1. Initial program 40.6%

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

      1. associate-/r* 40.6%

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

      2. associate-*l* 40.6%

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

      3. sqr-neg 40.6%

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

      4. associate-*l* 40.6%

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

      5. *-commutative 40.6%

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

      6. sqr-neg 40.6%

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

      7. associate-*l/ 40.6%

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

      8. associate-*r/ 40.5%

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

      9. associate-/r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in k around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r* 75.3%

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

      3. times-frac 75.4%

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

      4. unpow2 75.4%

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

      5. associate-*l* 78.6%

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

      6. unpow2 78.6%

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

   6. Simplified 78.6%

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

      1. frac-times 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in k around inf 75.4%

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

       1. associate-*r* 75.3%

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

       2. unpow2 75.3%

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

       3. associate-*r* 78.6%

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

       4. times-frac 78.6%

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

       5. associate-*r* 75.4%

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

       6. unpow2 75.4%

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

       7. unpow2 75.4%

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

       8. *-commutative 75.4%

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

       9. times-frac 85.8%

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

       10. unpow2 85.8%

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

   11. Simplified 85.8%

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

   if 1.89999999999999997e-33 < t

   1. Initial program 62.2%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   3. Simplified 62.2%

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

      1. add-sqr-sqrt 62.2%

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

      2. pow2 62.2%

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

      3. sqrt-div 62.2%

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

      4. sqrt-pow1 69.5%

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

      5. metadata-eval 69.5%

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

      6. sqrt-prod 25.5%

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

      7. add-sqr-sqrt 77.2%

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

   5. Applied egg-rr 77.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\tan k \cdot {t}^{3}} \cdot \frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \end{array} \]

Alternative 10: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 8.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{2 + \frac{k \cdot k}{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-81)
   (/
    (/ 2.0 (/ (* k (/ (pow t 3.0) l)) (/ l k)))
    (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
   (if (<= k 2.9e-27)
     (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))
     (if (<= k 8.3e-16)
       (/
        (/ (/ 2.0 (tan k)) (/ k (* l (/ l (pow t 3.0)))))
        (+ 2.0 (/ (* k k) (* t t))))
       (* 2.0 (* (* l (/ l (* k k))) (/ (cos k) (* t (pow (sin k) 2.0)))))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-81) {
		tmp = (2.0 / ((k * (pow(t, 3.0) / l)) / (l / k))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else if (k <= 2.9e-27) {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	} else if (k <= 8.3e-16) {
		tmp = ((2.0 / tan(k)) / (k / (l * (l / pow(t, 3.0))))) / (2.0 + ((k * k) / (t * t)));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 5d-81) then
        tmp = (2.0d0 / ((k * ((t ** 3.0d0) / l)) / (l / k))) / (1.0d0 + (((k / t) ** 2.0d0) + 1.0d0))
    else if (k <= 2.9d-27) then
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    else if (k <= 8.3d-16) then
        tmp = ((2.0d0 / tan(k)) / (k / (l * (l / (t ** 3.0d0))))) / (2.0d0 + ((k * k) / (t * t)))
    else
        tmp = 2.0d0 * ((l * (l / (k * k))) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-81) {
		tmp = (2.0 / ((k * (Math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else if (k <= 2.9e-27) {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	} else if (k <= 8.3e-16) {
		tmp = ((2.0 / Math.tan(k)) / (k / (l * (l / Math.pow(t, 3.0))))) / (2.0 + ((k * k) / (t * t)));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5e-81:
		tmp = (2.0 / ((k * (math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (math.pow((k / t), 2.0) + 1.0))
	elif k <= 2.9e-27:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	elif k <= 8.3e-16:
		tmp = ((2.0 / math.tan(k)) / (k / (l * (l / math.pow(t, 3.0))))) / (2.0 + ((k * k) / (t * t)))
	else:
		tmp = 2.0 * ((l * (l / (k * k))) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-81)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * Float64((t ^ 3.0) / l)) / Float64(l / k))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	elseif (k <= 2.9e-27)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	elseif (k <= 8.3e-16)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(k / Float64(l * Float64(l / (t ^ 3.0))))) / Float64(2.0 + Float64(Float64(k * k) / Float64(t * t))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-81)
		tmp = (2.0 / ((k * ((t ^ 3.0) / l)) / (l / k))) / (1.0 + (((k / t) ^ 2.0) + 1.0));
	elseif (k <= 2.9e-27)
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	elseif (k <= 8.3e-16)
		tmp = ((2.0 / tan(k)) / (k / (l * (l / (t ^ 3.0))))) / (2.0 + ((k * k) / (t * t)));
	else
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-81], N[(N[(2.0 / N[(N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.9e-27], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.3e-16], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 4.99999999999999981e-81

   1. Initial program 59.1%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. add-cube-cbrt 59.0%

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

      2. pow3 59.0%

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

      3. cbrt-div 59.0%

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

      4. rem-cbrt-cube 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in k around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. times-frac 56.4%

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

      4. unpow2 56.4%

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

      5. associate-/l* 59.8%

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

   8. Simplified 59.8%

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

      1. associate-*r/ 64.9%

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

   10. Applied egg-rr 64.9%

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

   if 4.99999999999999981e-81 < k < 2.90000000000000004e-27

   1. Initial program 56.7%

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

      1. associate-/r* 56.7%

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

      2. associate-*l* 56.7%

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

      3. sqr-neg 56.7%

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

      4. associate-*l* 56.7%

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

      5. *-commutative 56.7%

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

      6. sqr-neg 56.7%

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

      7. associate-*l/ 56.6%

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

      8. associate-*r/ 56.7%

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

      9. associate-/r/ 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in k around inf 31.1%

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

      1. *-commutative 31.1%

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

      2. associate-*r* 31.1%

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

      3. times-frac 31.1%

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

      4. unpow2 31.1%

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

      5. associate-*l* 31.1%

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

      6. unpow2 31.1%

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

   6. Simplified 31.1%

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

   7. Taylor expanded in k around 0 31.1%

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

      1. unpow2 31.1%

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

      2. *-commutative 31.1%

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

      3. times-frac 55.3%

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

   9. Simplified 55.3%

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

       1. associate-*r/ 68.6%

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

   11. Applied egg-rr 68.6%

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

   if 2.90000000000000004e-27 < k < 8.29999999999999978e-16

   1. Initial program 40.0%

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

      1. associate-/r* 40.0%

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

      2. associate-*l* 40.0%

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

      3. sqr-neg 40.0%

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

      4. associate-*l* 40.0%

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

      5. *-commutative 40.0%

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

      6. sqr-neg 40.0%

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

      7. associate-/r* 40.0%

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

   3. Simplified 20.0%

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

      1. expm1-log1p-u 20.0%

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

      2. expm1-udef 20.0%

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

      3. associate-/l* 20.0%

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

   5. Applied egg-rr 20.0%

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

      1. expm1-def 20.0%

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

      2. expm1-log1p 20.0%

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

      3. associate-/r/ 20.0%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in k around 0 40.0%

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

      1. associate-/l* 40.0%

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

      2. unpow2 40.0%

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

      3. associate-*r/ 43.9%

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

   10. Simplified 43.9%

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

   11. Taylor expanded in k around 0 43.9%

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

       1. unpow2 43.9%

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

       2. unpow2 43.9%

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

   13. Simplified 43.9%

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

   if 8.29999999999999978e-16 < k

   1. Initial program 42.8%

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

      1. associate-/r* 42.8%

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

      2. associate-*l* 42.8%

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

      3. sqr-neg 42.8%

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

      4. associate-*l* 42.8%

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

      5. *-commutative 42.8%

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

      6. sqr-neg 42.8%

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

      7. associate-/r* 42.8%

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

   3. Simplified 42.8%

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

      1. associate-/l/ 42.8%

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

      2. associate-/r/ 42.8%

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

      3. add-cube-cbrt 42.8%

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

      4. pow3 42.8%

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

   5. Applied egg-rr 55.4%

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

   6. Taylor expanded in k around inf 71.0%

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

      1. associate-*r/ 71.0%

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

      2. associate-*r* 71.0%

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

      3. unpow2 71.0%

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

      4. associate-*r* 73.6%

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

      5. associate-*r/ 73.6%

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

      6. associate-*r* 71.0%

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

      7. unpow2 71.0%

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

      8. associate-*r* 71.0%

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

      9. times-frac 67.2%

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

      10. unpow2 67.2%

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

      11. associate-*r/ 70.4%

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

      12. unpow2 70.4%

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

   8. Simplified 70.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 8.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\frac{k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{2 + \frac{k \cdot k}{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 11: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{t_1 \cdot \left(k \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 4e-81)
     (/
      (/ 2.0 (/ (* k (/ (pow t 3.0) l)) (/ l k)))
      (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
     (if (<= k 2.4e+131)
       (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) t_1)))
       (* 2.0 (* (/ (cos k) k) (/ (* l l) (* t_1 (* k t)))))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 4e-81) {
		tmp = (2.0 / ((k * (pow(t, 3.0) / l)) / (l / k))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else if (k <= 2.4e+131) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / t_1));
	} else {
		tmp = 2.0 * ((cos(k) / k) * ((l * l) / (t_1 * (k * t))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 4d-81) then
        tmp = (2.0d0 / ((k * ((t ** 3.0d0) / l)) / (l / k))) / (1.0d0 + (((k / t) ** 2.0d0) + 1.0d0))
    else if (k <= 2.4d+131) then
        tmp = 2.0d0 * (((l / t) * (l / (k * k))) * (cos(k) / t_1))
    else
        tmp = 2.0d0 * ((cos(k) / k) * ((l * l) / (t_1 * (k * t))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 4e-81) {
		tmp = (2.0 / ((k * (Math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else if (k <= 2.4e+131) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / t_1));
	} else {
		tmp = 2.0 * ((Math.cos(k) / k) * ((l * l) / (t_1 * (k * t))));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 4e-81:
		tmp = (2.0 / ((k * (math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (math.pow((k / t), 2.0) + 1.0))
	elif k <= 2.4e+131:
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (math.cos(k) / t_1))
	else:
		tmp = 2.0 * ((math.cos(k) / k) * ((l * l) / (t_1 * (k * t))))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 4e-81)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * Float64((t ^ 3.0) / l)) / Float64(l / k))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	elseif (k <= 2.4e+131)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / k) * Float64(Float64(l * l) / Float64(t_1 * Float64(k * t)))));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 4e-81)
		tmp = (2.0 / ((k * ((t ^ 3.0) / l)) / (l / k))) / (1.0 + (((k / t) ^ 2.0) + 1.0));
	elseif (k <= 2.4e+131)
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / t_1));
	else
		tmp = 2.0 * ((cos(k) / k) * ((l * l) / (t_1 * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 4e-81], N[(N[(2.0 / N[(N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.4e+131], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t$95$1 * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.9999999999999998e-81

   1. Initial program 59.1%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. add-cube-cbrt 59.0%

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

      2. pow3 59.0%

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

      3. cbrt-div 59.0%

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

      4. rem-cbrt-cube 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in k around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. times-frac 56.4%

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

      4. unpow2 56.4%

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

      5. associate-/l* 59.8%

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

   8. Simplified 59.8%

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

      1. associate-*r/ 64.9%

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

   10. Applied egg-rr 64.9%

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

   if 3.9999999999999998e-81 < k < 2.3999999999999999e131

   1. Initial program 43.0%

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

      1. associate-/r* 43.1%

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

      2. associate-*l* 43.1%

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

      3. sqr-neg 43.1%

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

      4. associate-*l* 43.1%

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

      5. *-commutative 43.1%

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

      6. sqr-neg 43.1%

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

      7. associate-*l/ 43.0%

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

      8. associate-*r/ 43.1%

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

      9. associate-/r/ 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in k around inf 64.1%

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

      1. *-commutative 64.1%

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

      2. associate-*r* 64.2%

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

      3. times-frac 64.1%

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

      4. unpow2 64.1%

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

      5. associate-*l* 64.1%

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

      6. unpow2 64.1%

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

   6. Simplified 64.1%

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

      1. frac-times 64.2%

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

   8. Applied egg-rr 64.2%

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

   9. Taylor expanded in k around inf 64.1%

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

       1. associate-*r* 64.2%

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

       2. unpow2 64.2%

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

       3. associate-*r* 64.2%

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

       4. times-frac 66.4%

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

       5. associate-*r* 66.4%

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

       6. unpow2 66.4%

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

       7. unpow2 66.4%

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

       8. *-commutative 66.4%

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

       9. times-frac 74.1%

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

       10. unpow2 74.1%

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

   11. Simplified 74.1%

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

   if 2.3999999999999999e131 < k

   1. Initial program 44.5%

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

      1. associate-/r* 44.5%

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

      2. associate-*l* 44.5%

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

      3. sqr-neg 44.5%

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

      4. associate-*l* 44.5%

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

      5. *-commutative 44.5%

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

      6. sqr-neg 44.5%

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

      7. associate-/r* 44.5%

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

   3. Simplified 44.5%

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

      1. associate-/l/ 44.5%

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

      2. associate-/r/ 44.5%

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

      3. add-cube-cbrt 44.5%

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

      4. pow3 44.5%

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

   5. Applied egg-rr 58.5%

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

   6. Taylor expanded in k around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. associate-*r* 65.1%

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

      3. unpow2 65.1%

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

      4. associate-*r* 69.4%

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

      5. associate-*r/ 69.4%

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

      6. *-commutative 69.4%

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

      7. associate-*l* 69.4%

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

      8. times-frac 74.9%

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

      9. unpow2 74.9%

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

   8. Simplified 74.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 12: 71.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 9e-81)
   (/
    (/ 2.0 (/ (* k (/ (pow t 3.0) l)) (/ l k)))
    (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
   (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (pow (sin k) 2.0))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e-81) {
		tmp = (2.0 / ((k * (pow(t, 3.0) / l)) / (l / k))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 9d-81) then
        tmp = (2.0d0 / ((k * ((t ** 3.0d0) / l)) / (l / k))) / (1.0d0 + (((k / t) ** 2.0d0) + 1.0d0))
    else
        tmp = 2.0d0 * (((l / t) * (l / (k * k))) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e-81) {
		tmp = (2.0 / ((k * (Math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 9e-81:
		tmp = (2.0 / ((k * (math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (math.pow((k / t), 2.0) + 1.0))
	else:
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 9e-81)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * Float64((t ^ 3.0) / l)) / Float64(l / k))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9e-81)
		tmp = (2.0 / ((k * ((t ^ 3.0) / l)) / (l / k))) / (1.0 + (((k / t) ^ 2.0) + 1.0));
	else
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 9e-81], N[(N[(2.0 / N[(N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.000000000000001e-81

   1. Initial program 59.1%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. add-cube-cbrt 59.0%

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

      2. pow3 59.0%

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

      3. cbrt-div 59.0%

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

      4. rem-cbrt-cube 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in k around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. times-frac 56.4%

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

      4. unpow2 56.4%

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

      5. associate-/l* 59.8%

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

   8. Simplified 59.8%

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

      1. associate-*r/ 64.9%

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

   10. Applied egg-rr 64.9%

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

   if 9.000000000000001e-81 < k

   1. Initial program 43.8%

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

      1. associate-/r* 43.8%

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

      2. associate-*l* 43.8%

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

      3. sqr-neg 43.8%

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

      4. associate-*l* 43.8%

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

      5. *-commutative 43.8%

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

      6. sqr-neg 43.8%

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

      7. associate-*l/ 43.8%

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

      8. associate-*r/ 43.8%

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

      9. associate-/r/ 43.7%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in k around inf 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 64.7%

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

      3. times-frac 64.7%

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

      4. unpow2 64.7%

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

      5. associate-*l* 66.9%

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

      6. unpow2 66.9%

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

   6. Simplified 66.9%

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

      1. frac-times 66.9%

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

   8. Applied egg-rr 66.9%

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

   9. Taylor expanded in k around inf 64.6%

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

       1. associate-*r* 64.7%

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

       2. unpow2 64.7%

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

       3. associate-*r* 66.9%

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

       4. times-frac 68.0%

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

       5. associate-*r* 65.7%

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

       6. unpow2 65.7%

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

       7. unpow2 65.7%

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

       8. *-commutative 65.7%

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

       9. times-frac 70.9%

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

       10. unpow2 70.9%

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

   11. Simplified 70.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 13: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{elif}\;k \leq 1.45:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6e-81)
   (/
    (/ 2.0 (/ (* k (/ (pow t 3.0) l)) (/ l k)))
    (+ 1.0 (+ (pow (/ k t) 2.0) 1.0)))
   (if (<= k 1.45)
     (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))
     (* 2.0 (* (* (/ l k) (/ l k)) (* (/ (cos k) t) 0.3333333333333333))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-81) {
		tmp = (2.0 / ((k * (pow(t, 3.0) / l)) / (l / k))) / (1.0 + (pow((k / t), 2.0) + 1.0));
	} else if (k <= 1.45) {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 6d-81) then
        tmp = (2.0d0 / ((k * ((t ** 3.0d0) / l)) / (l / k))) / (1.0d0 + (((k / t) ** 2.0d0) + 1.0d0))
    else if (k <= 1.45d0) then
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-81) {
		tmp = (2.0 / ((k * (Math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (Math.pow((k / t), 2.0) + 1.0));
	} else if (k <= 1.45) {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((Math.cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6e-81:
		tmp = (2.0 / ((k * (math.pow(t, 3.0) / l)) / (l / k))) / (1.0 + (math.pow((k / t), 2.0) + 1.0))
	elif k <= 1.45:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((math.cos(k) / t) * 0.3333333333333333))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6e-81)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * Float64((t ^ 3.0) / l)) / Float64(l / k))) / Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)));
	elseif (k <= 1.45)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(cos(k) / t) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6e-81)
		tmp = (2.0 / ((k * ((t ^ 3.0) / l)) / (l / k))) / (1.0 + (((k / t) ^ 2.0) + 1.0));
	elseif (k <= 1.45)
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6e-81], N[(N[(2.0 / N[(N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.45], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.9999999999999998e-81

   1. Initial program 59.1%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.0%

         \[\leadsto \frac{\frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      2. pow3 59.0%

         \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. cbrt-div 59.0%

         \[\leadsto \frac{\frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. rem-cbrt-cube 67.9%

         \[\leadsto \frac{\frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 67.9%

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   6. Taylor expanded in k around 0 52.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 52.5%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      2. unpow2 52.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. times-frac 56.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{{k}^{2}}{\ell}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. unpow2 56.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-/l* 59.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   8. Simplified 59.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   10. Applied egg-rr 64.9%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   if 5.9999999999999998e-81 < k < 1.44999999999999996

   1. Initial program 42.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 30.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 30.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 30.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 30.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 30.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 30.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 30.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 30.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 30.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 30.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 30.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 42.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 42.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 49.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   11. Applied egg-rr 49.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   if 1.44999999999999996 < k

   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* 44.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 44.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 44.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 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/ 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 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 71.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 71.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 71.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 71.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 74.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 74.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 74.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 60.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 60.6%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 60.6%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t} \cdot 0.3333333333333333\right)} \]

       2. times-frac 60.2%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)} \cdot 0.3333333333333333\right) \]

       3. unpow2 60.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       4. associate-*r/ 62.3%

          \[\leadsto 2 \cdot \left(\left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       5. unpow2 62.3%

          \[\leadsto 2 \cdot \left(\left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       6. associate-*l* 62.3%

          \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]

       7. associate-/r* 64.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       8. associate-*r/ 63.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell \cdot \frac{\ell}{k}}{k}} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       9. associate-*l/ 64.3%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

   12. Simplified 64.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{k}}}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\\ \mathbf{elif}\;k \leq 1.45:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 14: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq 4.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{t_1 \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.16 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l k))))
   (if (<= k 4.8e-145)
     (/ (* t_1 (/ 2.0 (pow t 3.0))) (+ 2.0 (pow (/ k t) 2.0)))
     (if (<= k 1.16e-14)
       (/ (* l (/ l (pow t 3.0))) (* k k))
       (* 2.0 (* t_1 (* (/ (cos k) t) 0.3333333333333333)))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double tmp;
	if (k <= 4.8e-145) {
		tmp = (t_1 * (2.0 / pow(t, 3.0))) / (2.0 + pow((k / t), 2.0));
	} else if (k <= 1.16e-14) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * (t_1 * ((cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) * (l / k)
    if (k <= 4.8d-145) then
        tmp = (t_1 * (2.0d0 / (t ** 3.0d0))) / (2.0d0 + ((k / t) ** 2.0d0))
    else if (k <= 1.16d-14) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * (t_1 * ((cos(k) / t) * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double tmp;
	if (k <= 4.8e-145) {
		tmp = (t_1 * (2.0 / Math.pow(t, 3.0))) / (2.0 + Math.pow((k / t), 2.0));
	} else if (k <= 1.16e-14) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * (t_1 * ((Math.cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	t_1 = (l / k) * (l / k)
	tmp = 0
	if k <= 4.8e-145:
		tmp = (t_1 * (2.0 / math.pow(t, 3.0))) / (2.0 + math.pow((k / t), 2.0))
	elif k <= 1.16e-14:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * (t_1 * ((math.cos(k) / t) * 0.3333333333333333))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / k))
	tmp = 0.0
	if (k <= 4.8e-145)
		tmp = Float64(Float64(t_1 * Float64(2.0 / (t ^ 3.0))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (k <= 1.16e-14)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(cos(k) / t) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) * (l / k);
	tmp = 0.0;
	if (k <= 4.8e-145)
		tmp = (t_1 * (2.0 / (t ^ 3.0))) / (2.0 + ((k / t) ^ 2.0));
	elseif (k <= 1.16e-14)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * (t_1 * ((cos(k) / t) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.8e-145], N[(N[(t$95$1 * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.16e-14], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.8000000000000003e-145

   1. Initial program 58.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 58.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 58.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 58.0%

         \[\leadsto \frac{\frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      2. pow3 58.0%

         \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. cbrt-div 58.0%

         \[\leadsto \frac{\frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. rem-cbrt-cube 67.8%

         \[\leadsto \frac{\frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 67.8%

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   6. Taylor expanded in k around 0 50.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      2. unpow2 50.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. times-frac 55.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{{k}^{2}}{\ell}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. unpow2 55.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-/l* 59.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   8. Simplified 59.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. div-inv 59.4%

         \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}} \cdot \frac{1}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. associate-*l/ 58.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}} \cdot \frac{1}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. associate-/r/ 58.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}}{\ell}} \cdot \frac{1}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. associate-+r+ 58.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}} \cdot \frac{1}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]

      5. metadata-eval 58.7%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}} \cdot \frac{1}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Applied egg-rr 58.7%

       \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 58.7%

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3} \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}} \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       2. *-rgt-identity 58.7%

          \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{t}^{3} \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       3. associate-/l* 58.7%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell}{\frac{k}{\ell} \cdot k}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       4. associate-*l/ 55.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell}{\color{blue}{\frac{k \cdot k}{\ell}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       5. unpow2 55.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell}{\frac{\color{blue}{{k}^{2}}}{\ell}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. associate-/l* 50.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell \cdot \ell}{{k}^{2}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       7. unpow2 50.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\color{blue}{{\ell}^{2}}}{{k}^{2}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       8. associate-/r/ 50.2%

          \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       9. unpow2 50.2%

          \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       10. unpow2 50.2%

           \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       11. times-frac 60.5%

           \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   12. Simplified 60.5%

       \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   if 4.8000000000000003e-145 < k < 1.16000000000000007e-14

   1. Initial program 60.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 60.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 60.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 60.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 60.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 60.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 60.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 60.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 60.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 64.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 64.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 64.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 62.7%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 62.7%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   if 1.16000000000000007e-14 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 71.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 71.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 71.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 71.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 73.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 73.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 73.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.6%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 60.6%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 59.1%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 59.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t} \cdot 0.3333333333333333\right)} \]

       2. times-frac 58.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)} \cdot 0.3333333333333333\right) \]

       3. unpow2 58.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       4. associate-*r/ 60.7%

          \[\leadsto 2 \cdot \left(\left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       5. unpow2 60.7%

          \[\leadsto 2 \cdot \left(\left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       6. associate-*l* 60.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]

       7. associate-/r* 62.4%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       8. associate-*r/ 62.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell \cdot \frac{\ell}{k}}{k}} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       9. associate-*l/ 62.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

   12. Simplified 62.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.16 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 15: 65.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}{1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;k \leq 1.45:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.1e-80)
   (/
    (/ 2.0 (* (/ (pow t 3.0) l) (/ k (/ l k))))
    (+ 1.0 (+ 1.0 (* (/ k t) (/ k t)))))
   (if (<= k 1.45)
     (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))
     (* 2.0 (* (* (/ l k) (/ l k)) (* (/ (cos k) t) 0.3333333333333333))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e-80) {
		tmp = (2.0 / ((pow(t, 3.0) / l) * (k / (l / k)))) / (1.0 + (1.0 + ((k / t) * (k / t))));
	} else if (k <= 1.45) {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 3.1d-80) then
        tmp = (2.0d0 / (((t ** 3.0d0) / l) * (k / (l / k)))) / (1.0d0 + (1.0d0 + ((k / t) * (k / t))))
    else if (k <= 1.45d0) then
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e-80) {
		tmp = (2.0 / ((Math.pow(t, 3.0) / l) * (k / (l / k)))) / (1.0 + (1.0 + ((k / t) * (k / t))));
	} else if (k <= 1.45) {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((Math.cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.1e-80:
		tmp = (2.0 / ((math.pow(t, 3.0) / l) * (k / (l / k)))) / (1.0 + (1.0 + ((k / t) * (k / t))))
	elif k <= 1.45:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((math.cos(k) / t) * 0.3333333333333333))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.1e-80)
		tmp = Float64(Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(k / Float64(l / k)))) / Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (k <= 1.45)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(cos(k) / t) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.1e-80)
		tmp = (2.0 / (((t ^ 3.0) / l) * (k / (l / k)))) / (1.0 + (1.0 + ((k / t) * (k / t))));
	elseif (k <= 1.45)
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.1e-80], N[(N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.45], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.10000000000000016e-80

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 59.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 59.0%

         \[\leadsto \frac{\frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      2. pow3 59.0%

         \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. cbrt-div 59.0%

         \[\leadsto \frac{\frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. rem-cbrt-cube 67.9%

         \[\leadsto \frac{\frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 67.9%

      \[\leadsto \frac{\frac{2}{\left(\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \tan k}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   6. Taylor expanded in k around 0 52.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 52.5%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      2. unpow2 52.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. times-frac 56.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{{k}^{2}}{\ell}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      4. unpow2 56.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-/l* 59.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   8. Simplified 59.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 59.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}{1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)} \]

   10. Applied egg-rr 59.8%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}{1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)} \]

   if 3.10000000000000016e-80 < k < 1.44999999999999996

   1. Initial program 42.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 30.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 30.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 30.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 30.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 30.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 30.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 30.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 30.5%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 30.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 30.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 30.7%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 42.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 42.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 49.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   11. Applied egg-rr 49.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   if 1.44999999999999996 < k

   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* 44.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 44.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 44.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 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/ 44.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 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 71.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 71.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 71.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 71.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 74.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 74.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 74.3%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 60.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 60.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 60.6%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 60.6%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t} \cdot 0.3333333333333333\right)} \]

       2. times-frac 60.2%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)} \cdot 0.3333333333333333\right) \]

       3. unpow2 60.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       4. associate-*r/ 62.3%

          \[\leadsto 2 \cdot \left(\left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       5. unpow2 62.3%

          \[\leadsto 2 \cdot \left(\left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       6. associate-*l* 62.3%

          \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]

       7. associate-/r* 64.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       8. associate-*r/ 63.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell \cdot \frac{\ell}{k}}{k}} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       9. associate-*l/ 64.3%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

   12. Simplified 64.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{k}{\frac{\ell}{k}}}}{1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;k \leq 1.45:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 16: 63.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.1e-14)
   (/ (* l (/ l (pow t 3.0))) (* k k))
   (* 2.0 (* (* (/ l k) (/ l k)) (* (/ (cos k) t) 0.3333333333333333)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.1e-14) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 2.1d-14) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.1e-14) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((Math.cos(k) / t) * 0.3333333333333333));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.1e-14:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((math.cos(k) / t) * 0.3333333333333333))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.1e-14)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(cos(k) / t) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.1e-14)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((cos(k) / t) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.1e-14], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.0999999999999999e-14

   1. Initial program 58.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 58.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 58.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 58.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 58.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 59.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 57.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 57.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 54.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 54.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 54.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 57.7%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 57.7%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 57.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   if 2.0999999999999999e-14 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 42.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 42.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 42.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 42.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 71.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 71.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 71.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 71.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 71.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 73.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 73.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 73.6%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 60.6%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. fma-def 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, \frac{{\ell}^{2}}{{k}^{2}}\right)}\right) \]

      2. unpow2 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\ell \cdot \ell}, \frac{{\ell}^{2}}{{k}^{2}}\right)\right) \]

      3. unpow2 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right)\right) \]

      4. associate-*r/ 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}\right)\right) \]

      5. unpow2 60.6%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right) \]

   9. Simplified 60.6%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \ell \cdot \ell, \ell \cdot \frac{\ell}{k \cdot k}\right)}\right) \]

   10. Taylor expanded in k around inf 59.1%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 59.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t} \cdot 0.3333333333333333\right)} \]

       2. times-frac 58.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)} \cdot 0.3333333333333333\right) \]

       3. unpow2 58.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       4. associate-*r/ 60.7%

          \[\leadsto 2 \cdot \left(\left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       5. unpow2 60.7%

          \[\leadsto 2 \cdot \left(\left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k}{t}\right) \cdot 0.3333333333333333\right) \]

       6. associate-*l* 60.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]

       7. associate-/r* 62.4%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       8. associate-*r/ 62.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell \cdot \frac{\ell}{k}}{k}} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

       9. associate-*l/ 62.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right) \]

   12. Simplified 62.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\cos k}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 17: 59.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.2e-80)
   (* (/ l (* k k)) (/ l (pow t 3.0)))
   (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.2e-80) {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 <= 3.2d-80) then
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.2e-80) {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.2e-80:
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.2e-80)
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.2e-80)
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.2e-80], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.1999999999999999e-80

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 59.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 58.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 58.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 54.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 54.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 54.4%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 58.1%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 58.1%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   if 3.1999999999999999e-80 < k

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 43.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 43.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 43.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 43.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 43.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 43.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 43.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 43.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 43.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 64.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 64.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 64.7%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 64.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 64.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 66.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 66.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 66.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 49.3%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 49.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 49.3%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 52.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 52.8%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 53.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   11. Applied egg-rr 53.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 18: 59.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-79)
   (/ (* l (/ l (pow t 3.0))) (* k k))
   (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-79) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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-79) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-79) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.5e-79:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-79)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-79)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.5e-79], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5e-79

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 58.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 58.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 58.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 58.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 59.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 58.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 58.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 54.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 54.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 54.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 57.7%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. unpow2 57.7%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 57.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   if 2.5e-79 < k

   1. Initial program 44.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 44.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 44.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 44.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 44.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 44.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 44.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 44.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 44.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 44.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 65.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 65.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 65.4%

         \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 65.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

      4. unpow2 65.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      5. associate-*l* 67.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

      6. unpow2 67.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

   6. Simplified 67.7%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 49.9%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 49.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 49.9%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 53.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 53.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 54.4%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   11. Applied egg-rr 54.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 19: 57.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

l = abs(l)
k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 54.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 54.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 54.2%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 54.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 53.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 63.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. *-commutative 63.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*r* 63.5%

      \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 63.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

   4. unpow2 63.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   5. associate-*l* 65.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   6. unpow2 65.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

6. Simplified 65.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 53.6%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 53.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 53.6%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 57.2%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 57.2%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Final simplification 57.2%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Alternative 20: 56.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (/ l (pow k 4.0))) t)))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l * (l / pow(k, 4.0))) / t);
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	return 2.0 * ((l * (l / math.pow(k, 4.0))) / t)

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t))
end

MATLAB

l = abs(l)
k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 54.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 54.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 54.2%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 54.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 53.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 63.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. *-commutative 63.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*r* 63.5%

      \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 63.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

   4. unpow2 63.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   5. associate-*l* 65.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   6. unpow2 65.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

6. Simplified 65.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 53.6%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 53.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 53.6%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 57.2%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 57.2%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
10. Step-by-step derivation

    1. associate-*l/ 57.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

11. Applied egg-rr 57.4%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

12. Final simplification 57.4%

    \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t} \]

Alternative 21: 56.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ 2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (/ l t)) (pow k 4.0))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l * (l / t)) / pow(k, 4.0));
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
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

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	return 2.0 * ((l * (l / t)) / math.pow(k, 4.0))

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)))
end

MATLAB

l = abs(l)
k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 54.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 54.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 54.2%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 54.5%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 53.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 63.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. *-commutative 63.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*r* 63.5%

      \[\leadsto 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 63.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)} \]

   4. unpow2 63.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   5. associate-*l* 65.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \]

   6. unpow2 65.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

6. Simplified 65.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 53.6%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 53.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 53.6%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 57.2%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 57.2%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
10. Step-by-step derivation

    1. associate-*r/ 57.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

11. Applied egg-rr 57.8%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

12. Final simplification 57.8%

    \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}} \]

Reproduce

herbie shell --seed 2023278 
(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))))