Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 91.5%

Time: 19.6s

Alternatives: 31

Speedup: 3.8×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% 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: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot t\_2}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2} \cdot t\_m}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\frac{t\_2}{\cos k}}\right)\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= (* l l) 1e-200)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+308)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) t_2)))
         k)
        (/
         (*
          (/
           (* (sqrt 2.0) t_m)
           (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (/ t_2 (cos k))))) 2.0))
          (*
           (/ (sqrt 2.0) k)
           (/ (pow (cbrt l) 2.0) (cbrt (* (sin k) (tan k))))))
         k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * t_2))) / k;
	} else {
		tmp = (((sqrt(2.0) * t_m) / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((t_2 / cos(k))))), 2.0)) * ((sqrt(2.0) / k) * (pow(cbrt(l), 2.0) / cbrt((sin(k) * tan(k)))))) / k;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * t_2))) / k;
	} else {
		tmp = (((Math.sqrt(2.0) * t_m) / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((t_2 / Math.cos(k))))), 2.0)) * ((Math.sqrt(2.0) / k) * (Math.pow(Math.cbrt(l), 2.0) / Math.cbrt((Math.sin(k) * Math.tan(k)))))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * t_2))) / k);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * t_m) / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(t_2 / cos(k))))) ^ 2.0)) * Float64(Float64(sqrt(2.0) / k) * Float64((cbrt(l) ^ 2.0) / cbrt(Float64(sin(k) * tan(k)))))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-/r* 45.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 52.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.2%

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

      2. associate-/r* 78.1%

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

      3. associate-/r/ 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/r/ 77.9%

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

      4. associate-*l/ 77.9%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.9%

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

       2. associate-*l/ 77.9%

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

       3. associate-*l/ 79.2%

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

       4. associate-*r/ 79.2%

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

   12. Simplified 80.6%

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

   13. Taylor expanded in l around -inf 98.4%

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

       1. *-commutative 98.4%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.4%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.4%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.4%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.7%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.8%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.8%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 34.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 34.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 84.4%

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

   8. Simplified 84.4%

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

      1. frac-times 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-/r/ 77.5%

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

      4. associate-*l/ 77.5%

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

      5. div-inv 77.5%

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

      6. pow-flip 77.5%

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

      7. metadata-eval 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. associate-/r* 77.5%

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

       2. associate-*l/ 77.5%

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

       3. associate-*l/ 84.4%

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

       4. associate-*r/ 84.5%

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

   12. Simplified 90.2%

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

   13. Taylor expanded in k around inf 90.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2} \cdot t}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\frac{{\sin k}^{2}}{\cos k}}\right)\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2} \cdot t\_m}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}}\right)}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 9.2e-101)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= l 9e+153)
      (/
       (*
        (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
        (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
       k)
      (/
       (*
        (/
         (* (sqrt 2.0) t_m)
         (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (* (sin k) (tan k))))) 2.0))
        (*
         (/ (sqrt 2.0) k)
         (/ (pow (cbrt l) 2.0) (* (cbrt (tan k)) (cbrt (sin k))))))
       k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 9.2e-101) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 9e+153) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = (((sqrt(2.0) * t_m) / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((sin(k) * tan(k))))), 2.0)) * ((sqrt(2.0) / k) * (pow(cbrt(l), 2.0) / (cbrt(tan(k)) * cbrt(sin(k)))))) / k;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 9.2e-101) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 9e+153) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = (((Math.sqrt(2.0) * t_m) / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((Math.sin(k) * Math.tan(k))))), 2.0)) * ((Math.sqrt(2.0) / k) * (Math.pow(Math.cbrt(l), 2.0) / (Math.cbrt(Math.tan(k)) * Math.cbrt(Math.sin(k)))))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 9.2e-101)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (l <= 9e+153)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * t_m) / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(sin(k) * tan(k))))) ^ 2.0)) * Float64(Float64(sqrt(2.0) / k) * Float64((cbrt(l) ^ 2.0) / Float64(cbrt(tan(k)) * cbrt(sin(k)))))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 9.2e-101], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 9e+153], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 9.1999999999999998e-101

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/r* 35.3%

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

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.0%

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

      2. associate-/l/ 30.0%

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

      3. associate-*l/ 29.5%

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

      4. associate-*l/ 30.1%

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

      5. *-commutative 30.1%

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

   8. Simplified 30.1%

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

   9. Taylor expanded in k around 0 37.5%

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

       1. associate-/l* 37.5%

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

   11. Simplified 37.5%

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

   if 9.1999999999999998e-101 < l < 9.0000000000000002e153

   1. Initial program 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/r* 45.0%

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

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 50.0%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 50.0%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 79.9%

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

      1. associate-/r/ 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r/ 81.6%

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

   8. Simplified 81.6%

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

      1. frac-times 81.4%

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

      2. associate-*l/ 81.4%

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

      3. associate-/r/ 81.5%

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

      4. associate-*l/ 81.5%

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

      5. div-inv 81.5%

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

      6. pow-flip 81.5%

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

      7. metadata-eval 81.5%

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

   10. Applied egg-rr 81.5%

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

       1. associate-/r* 81.5%

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

       2. associate-*l/ 81.5%

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

       3. associate-*l/ 81.7%

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

       4. associate-*r/ 81.6%

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

   12. Simplified 81.9%

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

   13. Taylor expanded in l around -inf 98.0%

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

       1. *-commutative 98.0%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.0%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.0%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.0%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.2%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.2%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.2%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 9.0000000000000002e153 < l

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-/r* 31.8%

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

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 31.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 31.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 86.0%

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

      1. associate-/r/ 86.0%

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

      2. associate-/r* 86.0%

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

      3. associate-/r/ 85.9%

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

   8. Simplified 85.9%

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

      1. frac-times 79.6%

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

      2. associate-*l/ 79.6%

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

      3. associate-/r/ 79.6%

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

      4. associate-*l/ 79.6%

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

      5. div-inv 79.6%

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

      6. pow-flip 79.6%

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

      7. metadata-eval 79.6%

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

   10. Applied egg-rr 79.6%

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

       1. associate-/r* 79.6%

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

       2. associate-*l/ 79.6%

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

       3. associate-*l/ 86.0%

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

       4. associate-*r/ 86.0%

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

   12. Simplified 88.5%

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

       1. *-commutative 88.5%

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

       2. cbrt-prod 88.6%

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

   14. Applied egg-rr 88.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2} \cdot t}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_2}\right) \cdot \left(\left(\sqrt{2} \cdot t\_m\right) \cdot {\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_2\right)\right)}^{-2}\right)}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k)))))
   (*
    t_s
    (if (<= (* l l) 1e-200)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 5e+277)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
         k)
        (/
         (*
          (* (/ (sqrt 2.0) k) (/ (pow (cbrt l) 2.0) t_2))
          (*
           (* (sqrt 2.0) t_m)
           (pow (* t_m (* (pow (cbrt l) -2.0) t_2)) -2.0)))
         k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 5e+277) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = (((sqrt(2.0) / k) * (pow(cbrt(l), 2.0) / t_2)) * ((sqrt(2.0) * t_m) * pow((t_m * (pow(cbrt(l), -2.0) * t_2)), -2.0))) / k;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 5e+277) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = (((Math.sqrt(2.0) / k) * (Math.pow(Math.cbrt(l), 2.0) / t_2)) * ((Math.sqrt(2.0) * t_m) * Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * t_2)), -2.0))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 5e+277)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) / k) * Float64((cbrt(l) ^ 2.0) / t_2)) * Float64(Float64(sqrt(2.0) * t_m) * (Float64(t_m * Float64((cbrt(l) ^ -2.0) * t_2)) ^ -2.0))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+277], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 4.99999999999999982e277

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

      2. associate-/r* 46.8%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 53.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 53.7%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.7%

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

      1. associate-/r/ 78.7%

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

      2. associate-/r* 78.7%

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

      3. associate-/r/ 79.8%

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

   8. Simplified 79.8%

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

      1. frac-times 78.5%

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

      2. associate-*l/ 78.4%

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

      3. associate-/r/ 78.5%

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

      4. associate-*l/ 78.5%

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

      5. div-inv 78.5%

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

      6. pow-flip 78.5%

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

      7. metadata-eval 78.5%

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

   10. Applied egg-rr 78.5%

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

       1. associate-/r* 78.5%

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

       2. associate-*l/ 78.5%

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

       3. associate-*l/ 79.8%

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

       4. associate-*r/ 79.8%

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

   12. Simplified 81.1%

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

   13. Taylor expanded in l around -inf 99.3%

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

       1. *-commutative 99.3%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 99.3%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 99.3%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 99.3%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 99.7%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 99.7%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 4.99999999999999982e277 < (*.f64 l l)

   1. Initial program 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/r* 33.5%

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

   3. Simplified 33.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.5%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 33.5%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 33.5%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 82.7%

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

      1. associate-/r/ 82.7%

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

      2. associate-/r* 82.7%

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

      3. associate-/r/ 83.5%

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

   8. Simplified 83.5%

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

      1. frac-times 76.8%

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

      2. associate-*l/ 76.8%

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

      3. associate-/r/ 76.8%

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

      4. associate-*l/ 76.8%

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

      5. div-inv 76.8%

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

      6. pow-flip 76.8%

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

      7. metadata-eval 76.8%

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

   10. Applied egg-rr 76.8%

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

       1. associate-/r* 76.8%

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

       2. associate-*l/ 76.8%

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

       3. associate-*l/ 83.6%

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

       4. associate-*r/ 83.6%

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

   12. Simplified 89.2%

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

       1. div-inv 88.2%

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

       2. pow-flip 89.2%

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

       3. metadata-eval 89.2%

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

   14. Applied egg-rr 89.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_2}\right) \cdot \left(t\_m \cdot \frac{\sqrt{2}}{{\left(t\_2 \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{2}}\right)}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k)))))
   (*
    t_s
    (if (<= (* l l) 1e-200)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+308)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
         k)
        (/
         (*
          (* (/ (sqrt 2.0) k) (/ (pow (cbrt l) 2.0) t_2))
          (* t_m (/ (sqrt 2.0) (pow (* t_2 (* t_m (pow (cbrt l) -2.0))) 2.0))))
         k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = (((sqrt(2.0) / k) * (pow(cbrt(l), 2.0) / t_2)) * (t_m * (sqrt(2.0) / pow((t_2 * (t_m * pow(cbrt(l), -2.0))), 2.0)))) / k;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = (((Math.sqrt(2.0) / k) * (Math.pow(Math.cbrt(l), 2.0) / t_2)) * (t_m * (Math.sqrt(2.0) / Math.pow((t_2 * (t_m * Math.pow(Math.cbrt(l), -2.0))), 2.0)))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) / k) * Float64((cbrt(l) ^ 2.0) / t_2)) * Float64(t_m * Float64(sqrt(2.0) / (Float64(t_2 * Float64(t_m * (cbrt(l) ^ -2.0))) ^ 2.0)))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[(t$95$2 * N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-/r* 45.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 52.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.2%

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

      2. associate-/r* 78.1%

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

      3. associate-/r/ 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/r/ 77.9%

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

      4. associate-*l/ 77.9%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.9%

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

       2. associate-*l/ 77.9%

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

       3. associate-*l/ 79.2%

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

       4. associate-*r/ 79.2%

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

   12. Simplified 80.6%

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

   13. Taylor expanded in l around -inf 98.4%

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

       1. *-commutative 98.4%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.4%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.4%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.4%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.7%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.8%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.8%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 34.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 34.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 84.4%

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

   8. Simplified 84.4%

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

      1. frac-times 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-/r/ 77.5%

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

      4. associate-*l/ 77.5%

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

      5. div-inv 77.5%

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

      6. pow-flip 77.5%

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

      7. metadata-eval 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. associate-/r* 77.5%

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

       2. associate-*l/ 77.5%

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

       3. associate-*l/ 84.4%

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

       4. associate-*r/ 84.5%

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

   12. Simplified 90.2%

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

       1. associate-*l/ 84.1%

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

       2. associate-*r/ 84.0%

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

   14. Applied egg-rr 84.0%

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

       1. associate-*l/ 90.2%

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

       2. *-commutative 90.2%

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

       3. associate-/l* 89.1%

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

       4. associate-*r* 89.1%

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

       5. *-commutative 89.1%

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

       6. associate-/l* 89.1%

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

   16. Simplified 89.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \cdot \left(t \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{2}}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k}\\ t_3 := \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot t\_2}{{\left(t\_3 \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{t\_2}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k)) (t_3 (cbrt (* (sin k) (tan k)))))
   (*
    t_s
    (if (<= (* l l) 1e-200)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+308)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
         k)
        (*
         (/ (* t_m t_2) (pow (* t_3 (/ t_m (pow (cbrt l) 2.0))) 2.0))
         (/ (/ t_2 (pow (cbrt l) -2.0)) t_3)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sqrt(2.0) / k;
	double t_3 = cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = ((t_m * t_2) / pow((t_3 * (t_m / pow(cbrt(l), 2.0))), 2.0)) * ((t_2 / pow(cbrt(l), -2.0)) / t_3);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sqrt(2.0) / k;
	double t_3 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = ((t_m * t_2) / Math.pow((t_3 * (t_m / Math.pow(Math.cbrt(l), 2.0))), 2.0)) * ((t_2 / Math.pow(Math.cbrt(l), -2.0)) / t_3);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sqrt(2.0) / k)
	t_3 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(Float64(t_m * t_2) / (Float64(t_3 * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 2.0)) * Float64(Float64(t_2 / (cbrt(l) ^ -2.0)) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Power[N[(t$95$3 * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-/r* 45.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 52.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.2%

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

      2. associate-/r* 78.1%

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

      3. associate-/r/ 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/r/ 77.9%

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

      4. associate-*l/ 77.9%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.9%

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

       2. associate-*l/ 77.9%

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

       3. associate-*l/ 79.2%

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

       4. associate-*r/ 79.2%

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

   12. Simplified 80.6%

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

   13. Taylor expanded in l around -inf 98.4%

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

       1. *-commutative 98.4%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.4%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.4%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.4%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.7%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.8%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.8%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 34.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 34.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 84.4%

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

   8. Simplified 84.4%

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

      1. *-un-lft-identity 84.4%

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

      2. associate-/l/ 84.4%

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

      3. associate-*l/ 84.4%

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

      4. *-commutative 84.4%

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

      5. div-inv 84.4%

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

      6. pow-flip 84.4%

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

      7. metadata-eval 84.4%

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

   10. Applied egg-rr 84.4%

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

       1. *-lft-identity 84.4%

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

       2. associate-/r* 84.4%

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

       3. associate-*l/ 84.4%

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

       4. associate-*r/ 84.4%

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

       5. *-commutative 84.4%

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

       6. associate-/r* 84.4%

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

       7. *-inverses 84.4%

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

       8. associate-*l/ 84.4%

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

       9. *-lft-identity 84.4%

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

   12. Simplified 84.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot t\_3}{{\left(t\_4 \cdot \frac{t\_m}{t\_2}\right)}^{2}} \cdot \frac{t\_3 \cdot t\_2}{t\_4}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (cbrt (* (sin k) (tan k)))))
   (*
    t_s
    (if (<= (* l l) 1e-200)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+308)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
         k)
        (*
         (/ (* t_m t_3) (pow (* t_4 (/ t_m t_2)) 2.0))
         (/ (* t_3 t_2) t_4)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(cbrt(l), 2.0);
	double t_3 = sqrt(2.0) / k;
	double t_4 = cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = ((t_m * t_3) / pow((t_4 * (t_m / t_2)), 2.0)) * ((t_3 * t_2) / t_4);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = ((t_m * t_3) / Math.pow((t_4 * (t_m / t_2)), 2.0)) * ((t_3 * t_2) / t_4);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(l) ^ 2.0
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(Float64(t_m * t_3) / (Float64(t_4 * Float64(t_m / t_2)) ^ 2.0)) * Float64(Float64(t_3 * t_2) / t_4));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[N[(t$95$4 * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * t$95$2), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-/r* 45.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 52.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.2%

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

      2. associate-/r* 78.1%

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

      3. associate-/r/ 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/r/ 77.9%

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

      4. associate-*l/ 77.9%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.9%

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

       2. associate-*l/ 77.9%

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

       3. associate-*l/ 79.2%

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

       4. associate-*r/ 79.2%

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

   12. Simplified 80.6%

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

   13. Taylor expanded in l around -inf 98.4%

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

       1. *-commutative 98.4%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.4%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.4%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.4%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.7%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.8%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.8%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 34.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 34.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 84.4%

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

   8. Simplified 84.4%

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

      1. *-un-lft-identity 84.4%

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

      2. associate-/r/ 84.4%

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

      3. associate-*l/ 84.4%

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

   10. Applied egg-rr 84.4%

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

       1. *-lft-identity 84.4%

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

       2. associate-/l/ 71.1%

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

       3. *-commutative 71.1%

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

       4. times-frac 84.4%

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

       5. *-inverses 84.4%

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

       6. *-lft-identity 84.4%

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

   12. Simplified 84.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_2}\right) \cdot \frac{\sqrt{2} \cdot t\_m}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_2\right)\right)}^{2}}}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k)))))
   (*
    t_s
    (if (<= l 9.2e-101)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= l 6.2e+150)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
         k)
        (/
         (*
          (* (/ (sqrt 2.0) k) (/ (pow (cbrt l) 2.0) t_2))
          (/ (* (sqrt 2.0) t_m) (pow (* t_m (* (pow (cbrt l) -2.0) t_2)) 2.0)))
         k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double tmp;
	if (l <= 9.2e-101) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 6.2e+150) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = (((sqrt(2.0) / k) * (pow(cbrt(l), 2.0) / t_2)) * ((sqrt(2.0) * t_m) / pow((t_m * (pow(cbrt(l), -2.0) * t_2)), 2.0))) / k;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (l <= 9.2e-101) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 6.2e+150) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = (((Math.sqrt(2.0) / k) * (Math.pow(Math.cbrt(l), 2.0) / t_2)) * ((Math.sqrt(2.0) * t_m) / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * t_2)), 2.0))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (l <= 9.2e-101)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (l <= 6.2e+150)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) / k) * Float64((cbrt(l) ^ 2.0) / t_2)) * Float64(Float64(sqrt(2.0) * t_m) / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * t_2)) ^ 2.0))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 9.2e-101], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 6.2e+150], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 9.1999999999999998e-101

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/r* 35.3%

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

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.0%

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

      2. associate-/l/ 30.0%

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

      3. associate-*l/ 29.5%

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

      4. associate-*l/ 30.1%

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

      5. *-commutative 30.1%

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

   8. Simplified 30.1%

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

   9. Taylor expanded in k around 0 37.5%

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

       1. associate-/l* 37.5%

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

   11. Simplified 37.5%

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

   if 9.1999999999999998e-101 < l < 6.20000000000000028e150

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

      2. associate-/r* 45.8%

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

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.9%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 50.9%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 50.9%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 81.3%

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

      1. associate-/r/ 81.3%

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

      2. associate-/r* 81.2%

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

      3. associate-/r/ 83.0%

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

   8. Simplified 83.0%

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

      1. frac-times 82.8%

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

      2. associate-*l/ 82.8%

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

      3. associate-/r/ 82.8%

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

      4. associate-*l/ 82.8%

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

      5. div-inv 82.9%

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

      6. pow-flip 82.9%

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

      7. metadata-eval 82.9%

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

   10. Applied egg-rr 82.9%

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

       1. associate-/r* 82.8%

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

       2. associate-*l/ 82.8%

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

       3. associate-*l/ 83.0%

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

       4. associate-*r/ 83.0%

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

   12. Simplified 83.2%

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

   13. Taylor expanded in l around -inf 99.6%

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

       1. *-commutative 99.6%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 99.5%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 99.5%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 99.5%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 99.8%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 99.8%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 6.20000000000000028e150 < l

   1. Initial program 31.1%

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

      1. *-commutative 31.1%

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

      2. associate-/r* 31.1%

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

   3. Simplified 31.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 31.1%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 31.1%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 84.1%

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

      1. associate-/r/ 84.1%

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

      2. associate-/r* 84.1%

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

      3. associate-/r/ 84.0%

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

   8. Simplified 84.0%

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

      1. frac-times 77.8%

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

      2. associate-*l/ 77.8%

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

      3. associate-/r/ 77.8%

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

      4. associate-*l/ 77.8%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.8%

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

       2. associate-*l/ 77.8%

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

       3. associate-*l/ 84.1%

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

       4. associate-*r/ 84.1%

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

   12. Simplified 86.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right) \cdot \frac{\sqrt{2} \cdot t}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.06 \cdot 10^{-100}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_m \cdot t\_3\right) \cdot \frac{\frac{\frac{t\_3}{t\_4}}{t\_2}}{{\left(t\_m \cdot \left(t\_4 \cdot t\_2\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= l 1.06e-100)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= l 1.35e+154)
        (/
         (*
          (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
          (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
         k)
        (*
         (* t_m t_3)
         (/ (/ (/ t_3 t_4) t_2) (pow (* t_m (* t_4 t_2)) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = pow(cbrt(l), -2.0);
	double tmp;
	if (l <= 1.06e-100) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 1.35e+154) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = (t_m * t_3) * (((t_3 / t_4) / t_2) / pow((t_m * (t_4 * t_2)), 2.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (l <= 1.06e-100) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 1.35e+154) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = (t_m * t_3) * (((t_3 / t_4) / t_2) / Math.pow((t_m * (t_4 * t_2)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (l <= 1.06e-100)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (l <= 1.35e+154)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(Float64(t_m * t_3) * Float64(Float64(Float64(t_3 / t_4) / t_2) / (Float64(t_m * Float64(t_4 * t_2)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 1.06e-100], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 1.35e+154], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(t$95$m * t$95$3), $MachinePrecision] * N[(N[(N[(t$95$3 / t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision] / N[Power[N[(t$95$m * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.0600000000000001e-100

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/r* 35.3%

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

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.0%

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

      2. associate-/l/ 30.0%

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

      3. associate-*l/ 29.5%

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

      4. associate-*l/ 30.1%

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

      5. *-commutative 30.1%

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

   8. Simplified 30.1%

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

   9. Taylor expanded in k around 0 37.5%

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

       1. associate-/l* 37.5%

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

   11. Simplified 37.5%

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

   if 1.0600000000000001e-100 < l < 1.35000000000000003e154

   1. Initial program 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/r* 45.0%

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

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 50.0%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 50.0%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 79.9%

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

      1. associate-/r/ 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r/ 81.6%

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

   8. Simplified 81.6%

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

      1. frac-times 81.4%

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

      2. associate-*l/ 81.4%

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

      3. associate-/r/ 81.5%

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

      4. associate-*l/ 81.5%

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

      5. div-inv 81.5%

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

      6. pow-flip 81.5%

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

      7. metadata-eval 81.5%

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

   10. Applied egg-rr 81.5%

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

       1. associate-/r* 81.5%

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

       2. associate-*l/ 81.5%

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

       3. associate-*l/ 81.7%

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

       4. associate-*r/ 81.6%

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

   12. Simplified 81.9%

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

   13. Taylor expanded in l around -inf 98.0%

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

       1. *-commutative 98.0%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.0%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.0%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.0%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.2%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.2%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.2%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 1.35000000000000003e154 < l

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-/r* 31.8%

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

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 31.8%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 31.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 86.0%

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

      1. associate-/r/ 86.0%

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

      2. associate-/r* 86.0%

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

      3. associate-/r/ 85.9%

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

   8. Simplified 85.9%

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

      1. associate-*l/ 79.6%

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

      2. associate-*l/ 79.6%

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

      3. associate-/l/ 79.5%

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

      4. associate-*l/ 79.6%

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

      5. *-commutative 79.6%

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

      6. div-inv 79.6%

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

      7. pow-flip 79.6%

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

      8. metadata-eval 79.6%

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

   10. Applied egg-rr 79.5%

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

       1. associate-/l* 86.0%

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

       2. associate-*l/ 86.0%

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

   12. Simplified 86.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.06 \cdot 10^{-100}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \frac{\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\frac{\sqrt{2}}{\sin k}}{k}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 9.2e-101)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= l 1.2e+154)
      (/
       (*
        (* 2.0 (* (pow (cbrt -1.0) 6.0) (cos k)))
        (/ (pow l 2.0) (* (* k t_m) (pow (sin k) 2.0))))
       k)
      (pow
       (* l (/ (/ (/ (sqrt 2.0) (sin k)) k) (sqrt (/ t_m (cos k)))))
       2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 9.2e-101) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 1.2e+154) {
		tmp = ((2.0 * (pow(cbrt(-1.0), 6.0) * cos(k))) * (pow(l, 2.0) / ((k * t_m) * pow(sin(k), 2.0)))) / k;
	} else {
		tmp = pow((l * (((sqrt(2.0) / sin(k)) / k) / sqrt((t_m / cos(k))))), 2.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 9.2e-101) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (l <= 1.2e+154) {
		tmp = ((2.0 * (Math.pow(Math.cbrt(-1.0), 6.0) * Math.cos(k))) * (Math.pow(l, 2.0) / ((k * t_m) * Math.pow(Math.sin(k), 2.0)))) / k;
	} else {
		tmp = Math.pow((l * (((Math.sqrt(2.0) / Math.sin(k)) / k) / Math.sqrt((t_m / Math.cos(k))))), 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 9.2e-101)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (l <= 1.2e+154)
		tmp = Float64(Float64(Float64(2.0 * Float64((cbrt(-1.0) ^ 6.0) * cos(k))) * Float64((l ^ 2.0) / Float64(Float64(k * t_m) * (sin(k) ^ 2.0)))) / k);
	else
		tmp = Float64(l * Float64(Float64(Float64(sqrt(2.0) / sin(k)) / k) / sqrt(Float64(t_m / cos(k))))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 9.2e-101], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 1.2e+154], N[(N[(N[(2.0 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[Power[N[(l * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 9.1999999999999998e-101

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/r* 35.3%

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

   3. Simplified 41.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.0%

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

      2. associate-/l/ 30.0%

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

      3. associate-*l/ 29.5%

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

      4. associate-*l/ 30.1%

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

      5. *-commutative 30.1%

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

   8. Simplified 30.1%

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

   9. Taylor expanded in k around 0 37.5%

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

       1. associate-/l* 37.5%

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

   11. Simplified 37.5%

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

   if 9.1999999999999998e-101 < l < 1.20000000000000007e154

   1. Initial program 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/r* 45.0%

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

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 50.0%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 50.0%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 79.9%

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

      1. associate-/r/ 79.9%

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

      2. associate-/r* 79.9%

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

      3. associate-/r/ 81.6%

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

   8. Simplified 81.6%

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

      1. frac-times 81.4%

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

      2. associate-*l/ 81.4%

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

      3. associate-/r/ 81.5%

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

      4. associate-*l/ 81.5%

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

      5. div-inv 81.5%

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

      6. pow-flip 81.5%

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

      7. metadata-eval 81.5%

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

   10. Applied egg-rr 81.5%

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

       1. associate-/r* 81.5%

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

       2. associate-*l/ 81.5%

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

       3. associate-*l/ 81.7%

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

       4. associate-*r/ 81.6%

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

   12. Simplified 81.9%

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

   13. Taylor expanded in l around -inf 98.0%

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

       1. *-commutative 98.0%

          \[\leadsto \frac{\frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot {\ell}^{2}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       2. associate-/l* 98.0%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}}{k} \]

       3. associate-*r* 98.0%

          \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)} \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       4. unpow2 98.0%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       5. rem-square-sqrt 98.2%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot \color{blue}{2}\right) \cdot \frac{{\ell}^{2}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}{k} \]

       6. associate-*r* 98.2%

          \[\leadsto \frac{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   15. Simplified 98.2%

       \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right) \cdot 2\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}}{k} \]

   if 1.20000000000000007e154 < l

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-/r* 31.8%

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

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 15.9%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 34.2%

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

      2. associate-/l/ 34.2%

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

      3. associate-*l/ 34.2%

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

      4. associate-*l/ 34.2%

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

      5. *-commutative 34.2%

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

   8. Simplified 34.2%

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

   9. Taylor expanded in k around inf 47.5%

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

       1. associate-*l/ 47.5%

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

       2. associate-/l* 47.5%

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

   11. Simplified 47.5%

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

       1. *-un-lft-identity 47.5%

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

       2. associate-*r/ 47.5%

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

   13. Applied egg-rr 47.5%

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

       1. *-lft-identity 47.5%

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

       2. associate-/r/ 47.5%

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

       3. associate-/r* 47.5%

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

       4. *-commutative 47.5%

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

       5. associate-/r* 47.6%

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

   15. Simplified 47.6%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \cos k\right)\right) \cdot \frac{{\ell}^{2}}{\left(k \cdot t\right) \cdot {\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\frac{\sqrt{2}}{\sin k}}{k}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t\_m} \cdot \frac{2}{{\sin k}^{2}}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\frac{\sqrt{2}}{\sin k}}{k}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-200)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 1e+308)
      (/ (* (pow l 2.0) (/ (* (/ (cos k) t_m) (/ 2.0 (pow (sin k) 2.0))) k)) k)
      (pow
       (* l (/ (/ (/ (sqrt 2.0) (sin k)) k) (sqrt (/ t_m (cos k)))))
       2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (pow(l, 2.0) * (((cos(k) / t_m) * (2.0 / pow(sin(k), 2.0))) / k)) / k;
	} else {
		tmp = pow((l * (((sqrt(2.0) / sin(k)) / k) / sqrt((t_m / cos(k))))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-200) then
        tmp = ((l * (sqrt(2.0d0) / (k ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 1d+308) then
        tmp = ((l ** 2.0d0) * (((cos(k) / t_m) * (2.0d0 / (sin(k) ** 2.0d0))) / k)) / k
    else
        tmp = (l * (((sqrt(2.0d0) / sin(k)) / k) / sqrt((t_m / cos(k))))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (Math.pow(l, 2.0) * (((Math.cos(k) / t_m) * (2.0 / Math.pow(Math.sin(k), 2.0))) / k)) / k;
	} else {
		tmp = Math.pow((l * (((Math.sqrt(2.0) / Math.sin(k)) / k) / Math.sqrt((t_m / Math.cos(k))))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-200:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 1e+308:
		tmp = (math.pow(l, 2.0) * (((math.cos(k) / t_m) * (2.0 / math.pow(math.sin(k), 2.0))) / k)) / k
	else:
		tmp = math.pow((l * (((math.sqrt(2.0) / math.sin(k)) / k) / math.sqrt((t_m / math.cos(k))))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64((l ^ 2.0) * Float64(Float64(Float64(cos(k) / t_m) * Float64(2.0 / (sin(k) ^ 2.0))) / k)) / k);
	else
		tmp = Float64(l * Float64(Float64(Float64(sqrt(2.0) / sin(k)) / k) / sqrt(Float64(t_m / cos(k))))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-200)
		tmp = ((l * (sqrt(2.0) / (k ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 1e+308)
		tmp = ((l ^ 2.0) * (((cos(k) / t_m) * (2.0 / (sin(k) ^ 2.0))) / k)) / k;
	else
		tmp = (l * (((sqrt(2.0) / sin(k)) / k) / sqrt((t_m / cos(k))))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[Power[N[(l * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-/r* 45.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 52.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.2%

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

      2. associate-/r* 78.1%

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

      3. associate-/r/ 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/r/ 77.9%

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

      4. associate-*l/ 77.9%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.9%

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

       2. associate-*l/ 77.9%

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

       3. associate-*l/ 79.2%

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

       4. associate-*r/ 79.2%

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

   12. Simplified 80.6%

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

   13. Taylor expanded in k around inf 80.6%

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

   14. Taylor expanded in t around 0 98.4%

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

       1. associate-/l* 98.4%

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

       2. *-commutative 98.4%

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

       3. associate-/r* 98.4%

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

       4. unpow2 98.4%

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

       5. rem-square-sqrt 98.7%

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

       6. times-frac 98.7%

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

   16. Simplified 98.7%

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

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 32.9%

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

      2. associate-/l/ 32.9%

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

      3. associate-*l/ 32.9%

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

      4. associate-*l/ 33.0%

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

      5. *-commutative 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in k around inf 49.7%

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

       1. associate-*l/ 49.7%

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

       2. associate-/l* 49.7%

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

   11. Simplified 49.7%

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

       1. *-un-lft-identity 49.7%

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

       2. associate-*r/ 49.7%

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

   13. Applied egg-rr 49.7%

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

       1. *-lft-identity 49.7%

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

       2. associate-/r/ 49.7%

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

       3. associate-/r* 49.7%

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

       4. *-commutative 49.7%

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

       5. associate-/r* 49.8%

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

   15. Simplified 49.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t} \cdot \frac{2}{{\sin k}^{2}}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\frac{\sqrt{2}}{\sin k}}{k}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t\_m} \cdot \frac{2}{{\sin k}^{2}}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-200)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 1e+308)
      (/ (* (pow l 2.0) (/ (* (/ (cos k) t_m) (/ 2.0 (pow (sin k) 2.0))) k)) k)
      (*
       2.0
       (pow (/ (/ 1.0 (/ (* k (sin k)) l)) (sqrt (/ t_m (cos k)))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (pow(l, 2.0) * (((cos(k) / t_m) * (2.0 / pow(sin(k), 2.0))) / k)) / k;
	} else {
		tmp = 2.0 * pow(((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-200) then
        tmp = ((l * (sqrt(2.0d0) / (k ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 1d+308) then
        tmp = ((l ** 2.0d0) * (((cos(k) / t_m) * (2.0d0 / (sin(k) ** 2.0d0))) / k)) / k
    else
        tmp = 2.0d0 * (((1.0d0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (Math.pow(l, 2.0) * (((Math.cos(k) / t_m) * (2.0 / Math.pow(Math.sin(k), 2.0))) / k)) / k;
	} else {
		tmp = 2.0 * Math.pow(((1.0 / ((k * Math.sin(k)) / l)) / Math.sqrt((t_m / Math.cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-200:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 1e+308:
		tmp = (math.pow(l, 2.0) * (((math.cos(k) / t_m) * (2.0 / math.pow(math.sin(k), 2.0))) / k)) / k
	else:
		tmp = 2.0 * math.pow(((1.0 / ((k * math.sin(k)) / l)) / math.sqrt((t_m / math.cos(k)))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64((l ^ 2.0) * Float64(Float64(Float64(cos(k) / t_m) * Float64(2.0 / (sin(k) ^ 2.0))) / k)) / k);
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(Float64(k * sin(k)) / l)) / sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-200)
		tmp = ((l * (sqrt(2.0) / (k ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 1e+308)
		tmp = ((l ^ 2.0) * (((cos(k) / t_m) * (2.0 / (sin(k) ^ 2.0))) / k)) / k;
	else
		tmp = 2.0 * (((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-/r* 45.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 52.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 52.7%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.2%

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

      2. associate-/r* 78.1%

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

      3. associate-/r/ 79.2%

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

   8. Simplified 79.2%

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

      1. frac-times 77.9%

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

      2. associate-*l/ 77.9%

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

      3. associate-/r/ 77.9%

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

      4. associate-*l/ 77.9%

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

      5. div-inv 77.9%

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

      6. pow-flip 77.9%

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

      7. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. associate-/r* 77.9%

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

       2. associate-*l/ 77.9%

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

       3. associate-*l/ 79.2%

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

       4. associate-*r/ 79.2%

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

   12. Simplified 80.6%

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

   13. Taylor expanded in k around inf 80.6%

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

   14. Taylor expanded in t around 0 98.4%

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

       1. associate-/l* 98.4%

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

       2. *-commutative 98.4%

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

       3. associate-/r* 98.4%

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

       4. unpow2 98.4%

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

       5. rem-square-sqrt 98.7%

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

       6. times-frac 98.7%

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

   16. Simplified 98.7%

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

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 32.9%

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

      2. associate-/l/ 32.9%

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

      3. associate-*l/ 32.9%

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

      4. associate-*l/ 33.0%

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

      5. *-commutative 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in k around inf 49.7%

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

       1. associate-*l/ 49.7%

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

       2. associate-/l* 49.7%

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

   11. Simplified 49.7%

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

       1. div-inv 49.7%

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

       2. unpow-prod-down 49.7%

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

       3. pow2 49.7%

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

       4. pow1/2 49.7%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 49.7%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 49.8%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 49.8%

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

       8. metadata-eval 49.8%

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

       9. associate-*r/ 49.8%

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

   13. Applied egg-rr 49.8%

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

       1. associate-*l/ 49.8%

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

       2. associate-/r* 49.8%

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

   15. Simplified 49.8%

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

Alternative 12: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-161}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(\frac{2}{t\_m \cdot {\sin k}^{2}} \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-161)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 1e+308)
      (/ (* (pow l 2.0) (* (/ 2.0 (* t_m (pow (sin k) 2.0))) (/ (cos k) k))) k)
      (*
       2.0
       (pow (/ (/ 1.0 (/ (* k (sin k)) l)) (sqrt (/ t_m (cos k)))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-161) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (pow(l, 2.0) * ((2.0 / (t_m * pow(sin(k), 2.0))) * (cos(k) / k))) / k;
	} else {
		tmp = 2.0 * pow(((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-161) then
        tmp = ((l * (sqrt(2.0d0) / (k ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 1d+308) then
        tmp = ((l ** 2.0d0) * ((2.0d0 / (t_m * (sin(k) ** 2.0d0))) * (cos(k) / k))) / k
    else
        tmp = 2.0d0 * (((1.0d0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-161) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (Math.pow(l, 2.0) * ((2.0 / (t_m * Math.pow(Math.sin(k), 2.0))) * (Math.cos(k) / k))) / k;
	} else {
		tmp = 2.0 * Math.pow(((1.0 / ((k * Math.sin(k)) / l)) / Math.sqrt((t_m / Math.cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-161:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 1e+308:
		tmp = (math.pow(l, 2.0) * ((2.0 / (t_m * math.pow(math.sin(k), 2.0))) * (math.cos(k) / k))) / k
	else:
		tmp = 2.0 * math.pow(((1.0 / ((k * math.sin(k)) / l)) / math.sqrt((t_m / math.cos(k)))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-161)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64((l ^ 2.0) * Float64(Float64(2.0 / Float64(t_m * (sin(k) ^ 2.0))) * Float64(cos(k) / k))) / k);
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(Float64(k * sin(k)) / l)) / sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-161)
		tmp = ((l * (sqrt(2.0) / (k ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 1e+308)
		tmp = ((l ^ 2.0) * ((2.0 / (t_m * (sin(k) ^ 2.0))) * (cos(k) / k))) / k;
	else
		tmp = 2.0 * (((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-161], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000000000000003e-161

   1. Initial program 29.4%

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

      1. *-commutative 29.4%

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

      2. associate-/r* 29.4%

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

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 27.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 27.6%

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

      2. associate-/l/ 27.5%

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

      3. associate-*l/ 26.7%

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

      4. associate-*l/ 27.8%

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

      5. *-commutative 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in k around 0 44.3%

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

       1. associate-/l* 44.3%

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

   11. Simplified 44.3%

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

   if 1.00000000000000003e-161 < (*.f64 l l) < 1e308

   1. Initial program 46.0%

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

      1. *-commutative 46.0%

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

      2. associate-/r* 46.1%

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

   3. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. add-cube-cbrt 53.4%

         \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

      3. times-frac 53.4%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 78.6%

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

      1. associate-/r/ 78.7%

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

      2. associate-/r* 78.6%

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

      3. associate-/r/ 79.8%

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

   8. Simplified 79.8%

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

      1. frac-times 78.4%

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

      2. associate-*l/ 78.4%

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

      3. associate-/r/ 78.4%

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

      4. associate-*l/ 78.4%

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

      5. div-inv 78.4%

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

      6. pow-flip 78.4%

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

      7. metadata-eval 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. associate-/r* 78.4%

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

       2. associate-*l/ 78.4%

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

       3. associate-*l/ 79.8%

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

       4. associate-*r/ 79.8%

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

   12. Simplified 80.1%

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

   13. Taylor expanded in t around 0 98.3%

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

       1. associate-/l* 98.3%

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

       2. *-commutative 98.3%

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

       3. unpow2 98.3%

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

       4. rem-square-sqrt 98.6%

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

       5. *-commutative 98.6%

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

       6. times-frac 98.6%

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

   15. Simplified 98.6%

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

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 32.9%

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

      2. associate-/l/ 32.9%

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

      3. associate-*l/ 32.9%

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

      4. associate-*l/ 33.0%

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

      5. *-commutative 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in k around inf 49.7%

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

       1. associate-*l/ 49.7%

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

       2. associate-/l* 49.7%

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

   11. Simplified 49.7%

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

       1. div-inv 49.7%

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

       2. unpow-prod-down 49.7%

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

       3. pow2 49.7%

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

       4. pow1/2 49.7%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 49.7%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 49.8%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 49.8%

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

       8. metadata-eval 49.8%

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

       9. associate-*r/ 49.8%

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

   13. Applied egg-rr 49.8%

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

       1. associate-*l/ 49.8%

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

       2. associate-/r* 49.8%

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

   15. Simplified 49.8%

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

Alternative 13: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\cos k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{t\_2}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (cos k))))
   (*
    t_s
    (if (<= (* l l) 1e-200)
      (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 1e+308)
        (* (* l l) (/ 2.0 (* (pow k 2.0) (* (pow (sin k) 2.0) t_2))))
        (* 2.0 (pow (/ (/ 1.0 (/ (* k (sin k)) l)) (sqrt t_2)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / cos(k);
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (l * l) * (2.0 / (pow(k, 2.0) * (pow(sin(k), 2.0) * t_2)));
	} else {
		tmp = 2.0 * pow(((1.0 / ((k * sin(k)) / l)) / sqrt(t_2)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m / cos(k)
    if ((l * l) <= 1d-200) then
        tmp = ((l * (sqrt(2.0d0) / (k ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 1d+308) then
        tmp = (l * l) * (2.0d0 / ((k ** 2.0d0) * ((sin(k) ** 2.0d0) * t_2)))
    else
        tmp = 2.0d0 * (((1.0d0 / ((k * sin(k)) / l)) / sqrt(t_2)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / Math.cos(k);
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (l * l) * (2.0 / (Math.pow(k, 2.0) * (Math.pow(Math.sin(k), 2.0) * t_2)));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / ((k * Math.sin(k)) / l)) / Math.sqrt(t_2)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m / math.cos(k)
	tmp = 0
	if (l * l) <= 1e-200:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 1e+308:
		tmp = (l * l) * (2.0 / (math.pow(k, 2.0) * (math.pow(math.sin(k), 2.0) * t_2)))
	else:
		tmp = 2.0 * math.pow(((1.0 / ((k * math.sin(k)) / l)) / math.sqrt(t_2)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / cos(k))
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 2.0) * Float64((sin(k) ^ 2.0) * t_2))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(Float64(k * sin(k)) / l)) / sqrt(t_2)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m / cos(k);
	tmp = 0.0;
	if ((l * l) <= 1e-200)
		tmp = ((l * (sqrt(2.0) / (k ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 1e+308)
		tmp = (l * l) * (2.0 / ((k ^ 2.0) * ((sin(k) ^ 2.0) * t_2)));
	else
		tmp = 2.0 * (((1.0 / ((k * sin(k)) / l)) / sqrt(t_2)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in t around 0 89.4%

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

      1. associate-/l* 89.4%

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

      2. *-commutative 89.4%

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

      3. associate-/l* 89.4%

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

   7. Simplified 89.4%

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

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 32.9%

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

      2. associate-/l/ 32.9%

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

      3. associate-*l/ 32.9%

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

      4. associate-*l/ 33.0%

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

      5. *-commutative 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in k around inf 49.7%

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

       1. associate-*l/ 49.7%

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

       2. associate-/l* 49.7%

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

   11. Simplified 49.7%

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

       1. div-inv 49.7%

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

       2. unpow-prod-down 49.7%

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

       3. pow2 49.7%

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

       4. pow1/2 49.7%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 49.7%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 49.8%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 49.8%

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

       8. metadata-eval 49.8%

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

       9. associate-*r/ 49.8%

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

   13. Applied egg-rr 49.8%

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

       1. associate-*l/ 49.8%

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

       2. associate-/r* 49.8%

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

   15. Simplified 49.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-200)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 1e+308)
      (* (* l l) (* 2.0 (/ (cos k) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))))
      (*
       2.0
       (pow (/ (/ 1.0 (/ (* k (sin k)) l)) (sqrt (/ t_m (cos k)))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (l * l) * (2.0 * (cos(k) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 * pow(((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-200) then
        tmp = ((l * (sqrt(2.0d0) / (k ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 1d+308) then
        tmp = (l * l) * (2.0d0 * (cos(k) / ((k ** 2.0d0) * (t_m * (sin(k) ** 2.0d0)))))
    else
        tmp = 2.0d0 * (((1.0d0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (l * l) * (2.0 * (Math.cos(k) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / ((k * Math.sin(k)) / l)) / Math.sqrt((t_m / Math.cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-200:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 1e+308:
		tmp = (l * l) * (2.0 * (math.cos(k) / (math.pow(k, 2.0) * (t_m * math.pow(math.sin(k), 2.0)))))
	else:
		tmp = 2.0 * math.pow(((1.0 / ((k * math.sin(k)) / l)) / math.sqrt((t_m / math.cos(k)))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(cos(k) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(Float64(k * sin(k)) / l)) / sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-200)
		tmp = ((l * (sqrt(2.0) / (k ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 1e+308)
		tmp = (l * l) * (2.0 * (cos(k) / ((k ^ 2.0) * (t_m * (sin(k) ^ 2.0)))));
	else
		tmp = 2.0 * (((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

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

      2. associate-/l/ 28.7%

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

      3. associate-*l/ 27.8%

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

      4. associate-*l/ 29.0%

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

      5. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. associate-/l* 43.1%

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

   11. Simplified 43.1%

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

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in t around 0 89.4%

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

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/r* 34.4%

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

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 32.9%

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

      2. associate-/l/ 32.9%

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

      3. associate-*l/ 32.9%

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

      4. associate-*l/ 33.0%

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

      5. *-commutative 33.0%

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

   8. Simplified 33.0%

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

   9. Taylor expanded in k around inf 49.7%

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

       1. associate-*l/ 49.7%

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

       2. associate-/l* 49.7%

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

   11. Simplified 49.7%

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

       1. div-inv 49.7%

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

       2. unpow-prod-down 49.7%

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

       3. pow2 49.7%

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

       4. pow1/2 49.7%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 49.7%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 49.8%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 49.8%

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

       8. metadata-eval 49.8%

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

       9. associate-*r/ 49.8%

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

   13. Applied egg-rr 49.8%

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

       1. associate-*l/ 49.8%

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

       2. associate-/r* 49.8%

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

   15. Simplified 49.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \left(\frac{{k}^{-2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-200)
    (pow (* (* l (/ (sqrt 2.0) (pow k 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 1e+308)
      (*
       (* l l)
       (* 2.0 (* (/ (pow k -2.0) t_m) (/ (cos k) (pow (sin k) 2.0)))))
      (*
       2.0
       (pow (/ (/ 1.0 (/ (* k (sin k)) l)) (sqrt (/ t_m (cos k)))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = pow(((l * (sqrt(2.0) / pow(k, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (l * l) * (2.0 * ((pow(k, -2.0) / t_m) * (cos(k) / pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 * pow(((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-200) then
        tmp = ((l * (sqrt(2.0d0) / (k ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 1d+308) then
        tmp = (l * l) * (2.0d0 * (((k ** (-2.0d0)) / t_m) * (cos(k) / (sin(k) ** 2.0d0))))
    else
        tmp = 2.0d0 * (((1.0d0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-200) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 1e+308) {
		tmp = (l * l) * (2.0 * ((Math.pow(k, -2.0) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / ((k * Math.sin(k)) / l)) / Math.sqrt((t_m / Math.cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-200:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 1e+308:
		tmp = (l * l) * (2.0 * ((math.pow(k, -2.0) / t_m) * (math.cos(k) / math.pow(math.sin(k), 2.0))))
	else:
		tmp = 2.0 * math.pow(((1.0 / ((k * math.sin(k)) / l)) / math.sqrt((t_m / math.cos(k)))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-200)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+308)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64((k ^ -2.0) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(Float64(k * sin(k)) / l)) / sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-200)
		tmp = ((l * (sqrt(2.0) / (k ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 1e+308)
		tmp = (l * l) * (2.0 * (((k ^ -2.0) / t_m) * (cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = 2.0 * (((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-200], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+308], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Power[k, -2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999999999998e-201

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.8%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 28.7%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 27.8%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 29.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 29.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 29.0%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 43.1%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 43.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)} \cdot \sqrt{\frac{1}{t}}\right)}^{2} \]

   11. Simplified 43.1%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 9.9999999999999998e-201 < (*.f64 l l) < 1e308

   1. Initial program 45.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 89.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. associate-/r* 89.4%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 89.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. div-inv 89.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

      2. div-inv 89.3%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\cos k \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right) \]

      3. pow-flip 89.3%

         \[\leadsto \left(2 \cdot \left(\left(\cos k \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right) \]

      4. metadata-eval 89.3%

         \[\leadsto \left(2 \cdot \left(\left(\cos k \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 89.3%

      \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos k \cdot {k}^{-2}\right) \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 89.4%

          \[\leadsto \left(2 \cdot \color{blue}{\frac{\left(\cos k \cdot {k}^{-2}\right) \cdot 1}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

       2. *-rgt-identity 89.4%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{\cos k \cdot {k}^{-2}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       3. *-commutative 89.4%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{{k}^{-2} \cdot \cos k}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       4. times-frac 89.4%

          \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{{k}^{-2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 89.4%

       \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{{k}^{-2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   if 1e308 < (*.f64 l l)

   1. Initial program 34.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 32.9%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 32.9%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 32.9%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 33.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 33.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 33.0%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around inf 49.7%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]
   10. Step-by-step derivation

       1. associate-*l/ 49.7%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

       2. associate-/l* 49.7%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   11. Simplified 49.7%

       \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]
   12. Step-by-step derivation

       1. div-inv 49.7%

          \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot \frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}}^{2} \]

       2. unpow-prod-down 49.7%

          \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]

       3. pow2 49.7%

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       4. pow1/2 49.7%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 49.7%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 49.8%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 49.8%

          \[\leadsto {2}^{\color{blue}{1}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       8. metadata-eval 49.8%

          \[\leadsto \color{blue}{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       9. associate-*r/ 49.8%

          \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   13. Applied egg-rr 49.8%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{1}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]
   14. Step-by-step derivation

       1. associate-*l/ 49.8%

          \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]

       2. associate-/r* 49.8%

          \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}}^{2} \]

   15. Simplified 49.8%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-200}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+308}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \left(\frac{{k}^{-2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+19}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t\_m}{\cos k}}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3e+19)
    (pow (/ (sqrt 2.0) (* (/ (pow k 2.0) l) (sqrt t_m))) 2.0)
    (* 2.0 (pow (/ (/ 1.0 (/ (* k (sin k)) l)) (sqrt (/ t_m (cos k)))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3e+19) {
		tmp = pow((sqrt(2.0) / ((pow(k, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * pow(((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3d+19) then
        tmp = (sqrt(2.0d0) / (((k ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((1.0d0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3e+19) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * Math.pow(((1.0 / ((k * Math.sin(k)) / l)) / Math.sqrt((t_m / Math.cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3e+19:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * math.pow(((1.0 / ((k * math.sin(k)) / l)) / math.sqrt((t_m / math.cos(k)))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3e+19)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(Float64(k * sin(k)) / l)) / sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3e+19)
		tmp = (sqrt(2.0) / (((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((1.0 / ((k * sin(k)) / l)) / sqrt((t_m / cos(k)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 3e+19], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3e19

   1. Initial program 37.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 37.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.9%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 29.4%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 29.4%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 28.9%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 29.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 29.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 37.5%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}}\right)}^{2} \]

   if 3e19 < l

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 36.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 23.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 35.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 35.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 35.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 35.1%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 35.1%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 35.1%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around inf 50.4%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]
   10. Step-by-step derivation

       1. associate-*l/ 50.4%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

       2. associate-/l* 50.3%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   11. Simplified 50.3%

       \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]
   12. Step-by-step derivation

       1. div-inv 50.3%

          \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot \frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}}^{2} \]

       2. unpow-prod-down 50.4%

          \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]

       3. pow2 50.4%

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       4. pow1/2 50.4%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 50.4%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 50.4%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 50.4%

          \[\leadsto {2}^{\color{blue}{1}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       8. metadata-eval 50.4%

          \[\leadsto \color{blue}{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       9. associate-*r/ 50.4%

          \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   13. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{1}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]
   14. Step-by-step derivation

       1. associate-*l/ 50.4%

          \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]

       2. associate-/r* 50.4%

          \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}}^{2} \]

   15. Simplified 50.4%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{\frac{1}{\frac{k \cdot \sin k}{\ell}}}{\sqrt{\frac{t}{\cos k}}}\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.35 \cdot 10^{+19}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\frac{\sqrt{\frac{t\_m}{\cos k}} \cdot \left(k \cdot \sin k\right)}{\ell}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3.35e+19)
    (pow (/ (sqrt 2.0) (* (/ (pow k 2.0) l) (sqrt t_m))) 2.0)
    (* 2.0 (pow (/ 1.0 (/ (* (sqrt (/ t_m (cos k))) (* k (sin k))) l)) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.35e+19) {
		tmp = pow((sqrt(2.0) / ((pow(k, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * pow((1.0 / ((sqrt((t_m / cos(k))) * (k * sin(k))) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.35d+19) then
        tmp = (sqrt(2.0d0) / (((k ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * ((1.0d0 / ((sqrt((t_m / cos(k))) * (k * sin(k))) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.35e+19) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * Math.pow((1.0 / ((Math.sqrt((t_m / Math.cos(k))) * (k * Math.sin(k))) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.35e+19:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * math.pow((1.0 / ((math.sqrt((t_m / math.cos(k))) * (k * math.sin(k))) / l)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.35e+19)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * (Float64(1.0 / Float64(Float64(sqrt(Float64(t_m / cos(k))) * Float64(k * sin(k))) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.35e+19)
		tmp = (sqrt(2.0) / (((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((1.0 / ((sqrt((t_m / cos(k))) * (k * sin(k))) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 3.35e+19], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[Power[N[(1.0 / N[(N[(N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.35e19

   1. Initial program 37.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 37.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.9%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 29.4%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 29.4%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 28.9%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 29.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 29.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 37.5%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}}\right)}^{2} \]

   if 3.35e19 < l

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 36.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 23.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 35.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 35.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 35.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 35.1%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 35.1%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 35.1%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around inf 50.4%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]
   10. Step-by-step derivation

       1. associate-*l/ 50.4%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

       2. associate-/l* 50.3%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   11. Simplified 50.3%

       \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]
   12. Step-by-step derivation

       1. div-inv 50.3%

          \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot \frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}}^{2} \]

       2. unpow-prod-down 50.4%

          \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]

       3. pow2 50.4%

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       4. pow1/2 50.4%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 50.4%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 50.4%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 50.4%

          \[\leadsto {2}^{\color{blue}{1}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       8. metadata-eval 50.4%

          \[\leadsto \color{blue}{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       9. associate-*r/ 50.4%

          \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   13. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{1}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.35 \cdot 10^{+19}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{\frac{\sqrt{\frac{t}{\cos k}} \cdot \left(k \cdot \sin k\right)}{\ell}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+19}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{1}{\sqrt{\frac{t\_m}{\cos k}} \cdot \left(k \cdot \sin k\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3.8e+19)
    (pow (/ (sqrt 2.0) (* (/ (pow k 2.0) l) (sqrt t_m))) 2.0)
    (* 2.0 (pow (* l (/ 1.0 (* (sqrt (/ t_m (cos k))) (* k (sin k))))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.8e+19) {
		tmp = pow((sqrt(2.0) / ((pow(k, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * pow((l * (1.0 / (sqrt((t_m / cos(k))) * (k * sin(k))))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.8d+19) then
        tmp = (sqrt(2.0d0) / (((k ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * ((l * (1.0d0 / (sqrt((t_m / cos(k))) * (k * sin(k))))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.8e+19) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * Math.pow((l * (1.0 / (Math.sqrt((t_m / Math.cos(k))) * (k * Math.sin(k))))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.8e+19:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * math.pow((l * (1.0 / (math.sqrt((t_m / math.cos(k))) * (k * math.sin(k))))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.8e+19)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * (Float64(l * Float64(1.0 / Float64(sqrt(Float64(t_m / cos(k))) * Float64(k * sin(k))))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.8e+19)
		tmp = (sqrt(2.0) / (((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((l * (1.0 / (sqrt((t_m / cos(k))) * (k * sin(k))))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 3.8e+19], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[Power[N[(l * N[(1.0 / N[(N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.8e19

   1. Initial program 37.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 37.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.9%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 29.4%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 29.4%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 28.9%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 29.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 29.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 37.5%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}}\right)}^{2} \]

   if 3.8e19 < l

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 36.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 23.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 35.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 35.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 35.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 35.1%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 35.1%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 35.1%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around inf 50.4%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]
   10. Step-by-step derivation

       1. associate-*l/ 50.4%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

       2. associate-/l* 50.3%

          \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   11. Simplified 50.3%

       \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]
   12. Step-by-step derivation

       1. div-inv 50.3%

          \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot \frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}}^{2} \]

       2. unpow-prod-down 50.4%

          \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]

       3. pow2 50.4%

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       4. pow1/2 50.4%

          \[\leadsto \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       5. pow1/2 50.4%

          \[\leadsto \left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       6. pow-prod-up 50.4%

          \[\leadsto \color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       7. metadata-eval 50.4%

          \[\leadsto {2}^{\color{blue}{1}} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       8. metadata-eval 50.4%

          \[\leadsto \color{blue}{2} \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2} \]

       9. associate-*r/ 50.4%

          \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

   13. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{1}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}\right)}^{2}} \]
   14. Step-by-step derivation

       1. associate-/r/ 50.4%

          \[\leadsto 2 \cdot {\color{blue}{\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}} \cdot \ell\right)}}^{2} \]

   15. Simplified 50.4%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{1}{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}} \cdot \ell\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+19}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{1}{\sqrt{\frac{t}{\cos k}} \cdot \left(k \cdot \sin k\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+172}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{{k}^{2} \cdot t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4.5e+172)
    (pow (/ (sqrt 2.0) (* (/ (pow k 2.0) l) (sqrt t_m))) 2.0)
    (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* (pow k 2.0) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.5e+172) {
		tmp = pow((sqrt(2.0) / ((pow(k, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (pow(k, 2.0) * t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.5d+172) then
        tmp = (sqrt(2.0d0) / (((k ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / ((k ** 2.0d0) * t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.5e+172) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (Math.pow(k, 2.0) * t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 4.5e+172:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (math.pow(k, 2.0) * t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4.5e+172)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64((k ^ 2.0) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 4.5e+172)
		tmp = (sqrt(2.0) / (((k ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / ((k ^ 2.0) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 4.5e+172], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.5000000000000002e172

   1. Initial program 37.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 37.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 43.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 30.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.8%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 30.8%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 30.5%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 30.9%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 30.9%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 38.1%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}}\right)}^{2} \]

   if 4.5000000000000002e172 < l

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 67.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. associate-/r* 67.7%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 67.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   8. Taylor expanded in k around 0 67.6%

      \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{\color{blue}{{k}^{2} \cdot t}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+172}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{{k}^{2} \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\frac{\sqrt{2}}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{t\_m}}{\ell}}\right)}^{2} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (pow (/ (sqrt 2.0) (* (* k (sin k)) (/ (sqrt t_m) l))) 2.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * pow((sqrt(2.0) / ((k * sin(k)) * (sqrt(t_m) / l))), 2.0);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((sqrt(2.0d0) / ((k * sin(k)) * (sqrt(t_m) / l))) ** 2.0d0)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * Math.pow((Math.sqrt(2.0) / ((k * Math.sin(k)) * (Math.sqrt(t_m) / l))), 2.0);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * math.pow((math.sqrt(2.0) / ((k * math.sin(k)) * (math.sqrt(t_m) / l))), 2.0)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(sqrt(2.0) / Float64(Float64(k * sin(k)) * Float64(sqrt(t_m) / l))) ^ 2.0))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((sqrt(2.0) / ((k * sin(k)) * (sqrt(t_m) / l))) ^ 2.0);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 37.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-/r* 37.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

3. Simplified 41.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 29.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

6. Applied egg-rr 31.1%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
7. Step-by-step derivation

   1. unpow2 31.1%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

   2. associate-/l/ 31.1%

      \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

   3. associate-*l/ 30.8%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

   4. associate-*l/ 31.2%

      \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

   5. *-commutative 31.2%

      \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

8. Simplified 31.2%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

9. Taylor expanded in k around inf 50.2%

   \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{2} \]
10. Step-by-step derivation

    1. associate-*l/ 50.2%

       \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

    2. associate-/l* 50.1%

       \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

11. Simplified 50.1%

    \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\left(k \cdot \sin k\right) \cdot \frac{\sqrt{\frac{t}{\cos k}}}{\ell}}}\right)}^{2} \]

12. Taylor expanded in k around 0 39.0%

    \[\leadsto {\left(\frac{\sqrt{2}}{\left(k \cdot \sin k\right) \cdot \color{blue}{\left(\frac{1}{\ell} \cdot \sqrt{t}\right)}}\right)}^{2} \]
13. Step-by-step derivation

    1. associate-*l/ 39.0%

       \[\leadsto {\left(\frac{\sqrt{2}}{\left(k \cdot \sin k\right) \cdot \color{blue}{\frac{1 \cdot \sqrt{t}}{\ell}}}\right)}^{2} \]

    2. *-lft-identity 39.0%

       \[\leadsto {\left(\frac{\sqrt{2}}{\left(k \cdot \sin k\right) \cdot \frac{\color{blue}{\sqrt{t}}}{\ell}}\right)}^{2} \]

14. Simplified 39.0%

    \[\leadsto {\left(\frac{\sqrt{2}}{\left(k \cdot \sin k\right) \cdot \color{blue}{\frac{\sqrt{t}}{\ell}}}\right)}^{2} \]
15. Add Preprocessing

Alternative 21: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.7 \cdot 10^{-136}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\left(k \cdot {t\_m}^{1.5}\right) \cdot \frac{k}{t\_m}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{{k}^{2} \cdot t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 7.7e-136)
    (pow (/ (sqrt 2.0) (/ (* (* k (pow t_m 1.5)) (/ k t_m)) l)) 2.0)
    (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* (pow k 2.0) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.7e-136) {
		tmp = pow((sqrt(2.0) / (((k * pow(t_m, 1.5)) * (k / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (pow(k, 2.0) * t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 7.7d-136) then
        tmp = (sqrt(2.0d0) / (((k * (t_m ** 1.5d0)) * (k / t_m)) / l)) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / ((k ** 2.0d0) * t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.7e-136) {
		tmp = Math.pow((Math.sqrt(2.0) / (((k * Math.pow(t_m, 1.5)) * (k / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (Math.pow(k, 2.0) * t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 7.7e-136:
		tmp = math.pow((math.sqrt(2.0) / (((k * math.pow(t_m, 1.5)) * (k / t_m)) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (math.pow(k, 2.0) * t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 7.7e-136)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(k * (t_m ^ 1.5)) * Float64(k / t_m)) / l)) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64((k ^ 2.0) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 7.7e-136)
		tmp = (sqrt(2.0) / (((k * (t_m ^ 1.5)) * (k / t_m)) / l)) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / ((k ^ 2.0) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 7.7e-136], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.6999999999999993e-136

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 34.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 29.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.6%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 28.6%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 28.0%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 28.7%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 28.7%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 28.7%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 28.7%

      \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\left(\color{blue}{k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   if 7.6999999999999993e-136 < l

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 81.8%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. associate-/r* 81.7%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 81.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   8. Taylor expanded in k around 0 71.4%

      \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{\color{blue}{{k}^{2} \cdot t}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.7 \cdot 10^{-136}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\left(k \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{{k}^{2} \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\left(k \cdot {t\_m}^{1.5}\right) \cdot \frac{k}{t\_m}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t\_m \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (pow (/ (sqrt 2.0) (/ (* (* k (pow t_m 1.5)) (/ k t_m)) l)) 2.0)
    (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t_m (pow (sin k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = pow((sqrt(2.0) / (((k * pow(t_m, 1.5)) * (k / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t_m * pow(sin(k), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = (sqrt(2.0d0) / (((k * (t_m ** 1.5d0)) * (k / t_m)) / l)) ** 2.0d0
    else
        tmp = (l * l) * ((2.0d0 * (k ** (-2.0d0))) / (t_m * (sin(k) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = Math.pow((Math.sqrt(2.0) / (((k * Math.pow(t_m, 1.5)) * (k / t_m)) / l)), 2.0);
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t_m * Math.pow(Math.sin(k), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = math.pow((math.sqrt(2.0) / (((k * math.pow(t_m, 1.5)) * (k / t_m)) / l)), 2.0)
	else:
		tmp = (l * l) * ((2.0 * math.pow(k, -2.0)) / (t_m * math.pow(math.sin(k), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(k * (t_m ^ 1.5)) * Float64(k / t_m)) / l)) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t_m * (sin(k) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (sqrt(2.0) / (((k * (t_m ^ 1.5)) * (k / t_m)) / l)) ^ 2.0;
	else
		tmp = (l * l) * ((2.0 * (k ^ -2.0)) / (t_m * (sin(k) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 24.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-/r* 24.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   3. Simplified 31.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 24.1%

         \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

      2. associate-/l/ 24.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}\right)}}^{2} \]

      3. associate-*l/ 23.8%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{{t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}}{\ell}} \cdot \frac{k}{t}}\right)}^{2} \]

      4. associate-*l/ 23.8%

         \[\leadsto {\left(\frac{\sqrt{2}}{\color{blue}{\frac{\left({t}^{1.5} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}}{\ell}}}\right)}^{2} \]

      5. *-commutative 23.8%

         \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right)} \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   8. Simplified 23.8%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{\left(\sqrt{\sin k \cdot \tan k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 33.1%

      \[\leadsto {\left(\frac{\sqrt{2}}{\frac{\left(\color{blue}{k} \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2} \]

   if 0.0 < (*.f64 l l)

   1. Initial program 40.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 45.4%

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. associate-/r* 79.0%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 79.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   8. Taylor expanded in k around 0 68.6%

      \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2}}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 68.6%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2}}}{t \cdot {\sin k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

      2. pow-flip 68.7%

         \[\leadsto \frac{2 \cdot \color{blue}{{k}^{\left(-2\right)}}}{t \cdot {\sin k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

      3. metadata-eval 68.7%

         \[\leadsto \frac{2 \cdot {k}^{\color{blue}{-2}}}{t \cdot {\sin k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-2}}{t \cdot {\sin k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\left(k \cdot {t}^{1.5}\right) \cdot \frac{k}{t}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t\_m \cdot {\sin k}^{2}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t_m (pow (sin k) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 * pow(k, -2.0)) / (t_m * pow(sin(k), 2.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * ((2.0d0 * (k ** (-2.0d0))) / (t_m * (sin(k) ** 2.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 * Math.pow(k, -2.0)) / (t_m * Math.pow(Math.sin(k), 2.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * ((2.0 * math.pow(k, -2.0)) / (t_m * math.pow(math.sin(k), 2.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t_m * (sin(k) ^ 2.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * ((2.0 * (k ^ -2.0)) / (t_m * (sin(k) ^ 2.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. associate-/r* 74.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in k around 0 66.4%

   \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2}}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
9. Step-by-step derivation

   1. associate-*r/ 66.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2}}}{t \cdot {\sin k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   2. pow-flip 66.5%

      \[\leadsto \frac{2 \cdot \color{blue}{{k}^{\left(-2\right)}}}{t \cdot {\sin k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   3. metadata-eval 66.5%

      \[\leadsto \frac{2 \cdot {k}^{\color{blue}{-2}}}{t \cdot {\sin k}^{2}} \cdot \left(\ell \cdot \ell\right) \]

10. Applied egg-rr 66.5%

    \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-2}}{t \cdot {\sin k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

11. Final simplification 66.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot {\sin k}^{2}} \]
12. Add Preprocessing

Alternative 24: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * (2.0d0 / ((k ** 2.0d0) * (t_m * (sin(k) ** 2.0d0)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (2.0 / (math.pow(k, 2.0) * (t_m * math.pow(math.sin(k), 2.0)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (2.0 / ((k ^ 2.0) * (t_m * (sin(k) ^ 2.0)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. associate-/r* 74.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in k around 0 66.4%

   \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2}}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Taylor expanded in k around inf 66.4%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \cdot \left(\ell \cdot \ell\right) \]

10. Final simplification 66.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]
11. Add Preprocessing

Alternative 25: 65.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{k} \cdot \frac{1}{k}}{t\_m \cdot {\sin k}^{2}}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (* (* l l) (* 2.0 (/ (* (/ 1.0 k) (/ 1.0 k)) (* t_m (pow (sin k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 * (((1.0 / k) * (1.0 / k)) / (t_m * pow(sin(k), 2.0)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * (2.0d0 * (((1.0d0 / k) * (1.0d0 / k)) / (t_m * (sin(k) ** 2.0d0)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 * (((1.0 / k) * (1.0 / k)) / (t_m * Math.pow(Math.sin(k), 2.0)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (2.0 * (((1.0 / k) * (1.0 / k)) / (t_m * math.pow(math.sin(k), 2.0)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(1.0 / k)) / Float64(t_m * (sin(k) ^ 2.0))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (2.0 * (((1.0 / k) * (1.0 / k)) / (t_m * (sin(k) ^ 2.0)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. associate-/r* 74.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in k around 0 66.4%

   \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2}}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
9. Step-by-step derivation

   1. inv-pow 66.4%

      \[\leadsto \left(2 \cdot \frac{\color{blue}{{\left({k}^{2}\right)}^{-1}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   2. unpow2 66.4%

      \[\leadsto \left(2 \cdot \frac{{\color{blue}{\left(k \cdot k\right)}}^{-1}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   3. unpow-prod-down 66.4%

      \[\leadsto \left(2 \cdot \frac{\color{blue}{{k}^{-1} \cdot {k}^{-1}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   4. inv-pow 66.4%

      \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{k}} \cdot {k}^{-1}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   5. inv-pow 66.4%

      \[\leadsto \left(2 \cdot \frac{\frac{1}{k} \cdot \color{blue}{\frac{1}{k}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

10. Applied egg-rr 66.4%

    \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{k} \cdot \frac{1}{k}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

11. Final simplification 66.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{k} \cdot \frac{1}{k}}{t \cdot {\sin k}^{2}}\right) \]
12. Add Preprocessing

Alternative 26: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{{k}^{2}}}{{k}^{2} \cdot t\_m}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (* 2.0 (/ (/ 1.0 (pow k 2.0)) (* (pow k 2.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 * ((1.0 / pow(k, 2.0)) / (pow(k, 2.0) * t_m))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * (2.0d0 * ((1.0d0 / (k ** 2.0d0)) / ((k ** 2.0d0) * t_m))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 * ((1.0 / Math.pow(k, 2.0)) / (Math.pow(k, 2.0) * t_m))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (2.0 * ((1.0 / math.pow(k, 2.0)) / (math.pow(k, 2.0) * t_m))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(1.0 / (k ^ 2.0)) / Float64((k ^ 2.0) * t_m)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (2.0 * ((1.0 / (k ^ 2.0)) / ((k ^ 2.0) * t_m))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. associate-/r* 74.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in k around 0 66.4%

   \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2}}}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Taylor expanded in k around 0 65.8%

   \[\leadsto \left(2 \cdot \frac{\frac{1}{{k}^{2}}}{\color{blue}{{k}^{2} \cdot t}}\right) \cdot \left(\ell \cdot \ell\right) \]

10. Final simplification 65.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{{k}^{2}}}{{k}^{2} \cdot t}\right) \]
11. Add Preprocessing

Alternative 27: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{3}}}{k} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* 2.0 (/ (pow l 2.0) (* t_m (pow k 3.0)))) k)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 * (pow(l, 2.0) / (t_m * pow(k, 3.0)))) / k);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 3.0d0)))) / k)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 3.0)))) / k);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 3.0)))) / k)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 3.0)))) / k))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 * ((l ^ 2.0) / (t_m * (k ^ 3.0)))) / k);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 37.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-/r* 37.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

3. Simplified 41.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 41.9%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. add-cube-cbrt 41.9%

      \[\leadsto \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}} \cdot \sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   3. times-frac 41.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\sqrt{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

6. Applied egg-rr 82.3%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
7. Step-by-step derivation

   1. associate-/r/ 82.3%

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   2. associate-/r* 82.3%

      \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

   3. associate-/r/ 83.0%

      \[\leadsto \frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\color{blue}{\frac{\sqrt{2}}{k} \cdot t}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

8. Simplified 83.0%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]
9. Step-by-step derivation

   1. frac-times 79.7%

      \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2}}{k} \cdot t\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]

   2. associate-*l/ 79.7%

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot t}{k}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   3. associate-/r/ 79.7%

      \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k} \cdot \color{blue}{\left(\frac{\frac{\sqrt{2}}{k} \cdot t}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   4. associate-*l/ 79.7%

      \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k} \cdot \left(\frac{\color{blue}{\frac{\sqrt{2} \cdot t}{k}}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   5. div-inv 79.7%

      \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\color{blue}{\left(t \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   6. pow-flip 79.7%

      \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\left(t \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

   7. metadata-eval 79.7%

      \[\leadsto \frac{\frac{\sqrt{2} \cdot t}{k} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \]

10. Applied egg-rr 79.7%

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2} \cdot t}{k} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}}} \]
11. Step-by-step derivation

    1. associate-/r* 79.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{2} \cdot t}{k} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k}}} \]

    2. associate-*l/ 79.7%

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

    3. associate-*l/ 83.0%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}}{\sqrt[3]{\sin k \cdot \tan k}} \]

    4. associate-*r/ 83.0%

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}\right)} \cdot \left(\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

12. Simplified 86.5%

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2} \cdot t}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}{k}} \]

13. Taylor expanded in k around 0 64.6%

    \[\leadsto \frac{\color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{3} \cdot t}}}{k} \]
14. Step-by-step derivation

    1. *-commutative 64.6%

       \[\leadsto \frac{\frac{\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot {\ell}^{2}}}{{k}^{3} \cdot t}}{k} \]

    2. unpow2 64.6%

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot {\ell}^{2}}{{k}^{3} \cdot t}}{k} \]

    3. rem-square-sqrt 64.6%

       \[\leadsto \frac{\frac{\color{blue}{2} \cdot {\ell}^{2}}{{k}^{3} \cdot t}}{k} \]

    4. associate-/l* 64.6%

       \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]

    5. *-commutative 64.6%

       \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{3}}}}{k} \]

15. Simplified 64.6%

    \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{3}}}}{k} \]
16. Add Preprocessing

Alternative 28: 62.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{{k}^{4}}}{t\_m}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (* 2.0 (/ (/ 1.0 (pow k 4.0)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 * ((1.0 / pow(k, 4.0)) / t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * (2.0d0 * ((1.0d0 / (k ** 4.0d0)) / t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 * ((1.0 / Math.pow(k, 4.0)) / t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (2.0 * ((1.0 / math.pow(k, 4.0)) / t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(1.0 / (k ^ 4.0)) / t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (2.0 * ((1.0 / (k ^ 4.0)) / t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(1.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. associate-/r* 74.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 74.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in k around 0 63.8%

   \[\leadsto \left(2 \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \cdot \left(\ell \cdot \ell\right) \]
9. Step-by-step derivation

   1. associate-/r* 63.8%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{1}{{k}^{4}}}{t}}\right) \cdot \left(\ell \cdot \ell\right) \]

10. Simplified 63.8%

    \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{1}{{k}^{4}}}{t}}\right) \cdot \left(\ell \cdot \ell\right) \]

11. Final simplification 63.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{{k}^{4}}}{t}\right) \]
12. Add Preprocessing

Alternative 29: 63.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 / (t_m * pow(k, 4.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * (2.0d0 / (t_m * (k ** 4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (2.0 / (t_m * Math.pow(k, 4.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (2.0 / (t_m * math.pow(k, 4.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (2.0 / (t_m * (k ^ 4.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 63.8%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 63.8%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 30: 63.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t\_m} \cdot {k}^{-4}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (* (/ 2.0 t_m) (pow k -4.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 / t_m) * pow(k, -4.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * ((2.0d0 / t_m) * (k ** (-4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * ((2.0 / t_m) * Math.pow(k, -4.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * ((2.0 / t_m) * math.pow(k, -4.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 / t_m) * (k ^ -4.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * ((2.0 / t_m) * (k ^ -4.0)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 63.8%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. *-commutative 63.8%

      \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   2. associate-/r* 63.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 63.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. div-inv 63.8%

      \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. pow-flip 63.8%

      \[\leadsto \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

   3. metadata-eval 63.8%

      \[\leadsto \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right)} \cdot \left(\ell \cdot \ell\right) \]

10. Final simplification 63.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right) \]
11. Add Preprocessing

Alternative 31: 20.5% accurate, 60.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l l) (/ -0.11666666666666667 t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) * ((-0.11666666666666667d0) / t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) * (-0.11666666666666667 / t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.11666666666666667 / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) * (-0.11666666666666667 / t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 46.0%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 17.7%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 17.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024174 
(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))))