Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 83.8%

Time: 25.0s

Alternatives: 19

Speedup: 3.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t_m}\right)\right)\\ t_3 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_4 := \sqrt[3]{\sqrt{l_m}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 1.45 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t_m}{\sqrt[3]{l_m}}}{\frac{1}{\sqrt[3]{\frac{\sin k}{l_m}}}}\right)}^{3} \cdot t_3}\\ \mathbf{elif}\;t_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)}}{\frac{t_2}{l_m}} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot {\left(\frac{\frac{t_m}{t_4 \cdot t_4}}{\frac{\sqrt[3]{l_m}}{\sqrt[3]{\sin k}}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k t_m))))
        (t_3 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))
        (t_4 (cbrt (sqrt l_m))))
   (*
    t_s
    (if (<= t_m 7.8e-159)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (<= t_m 1.45e-14)
        (/
         2.0
         (*
          (pow (/ (/ t_m (cbrt l_m)) (/ 1.0 (cbrt (/ (sin k) l_m)))) 3.0)
          t_3))
        (if (<= t_m 5e+102)
          (*
           (/ (/ 2.0 (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ t_2 l_m))
           (/ l_m t_2))
          (/
           2.0
           (*
            t_3
            (pow
             (/ (/ t_m (* t_4 t_4)) (/ (cbrt l_m) (cbrt (sin k))))
             3.0)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double t_3 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double t_4 = cbrt(sqrt(l_m));
	double tmp;
	if (t_m <= 7.8e-159) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if (t_m <= 1.45e-14) {
		tmp = 2.0 / (pow(((t_m / cbrt(l_m)) / (1.0 / cbrt((sin(k) / l_m)))), 3.0) * t_3);
	} else if (t_m <= 5e+102) {
		tmp = ((2.0 / (sin(k) * (tan(k) * pow(t_m, 3.0)))) / (t_2 / l_m)) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * pow(((t_m / (t_4 * t_4)) / (cbrt(l_m) / cbrt(sin(k)))), 3.0));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double t_3 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double t_4 = Math.cbrt(Math.sqrt(l_m));
	double tmp;
	if (t_m <= 7.8e-159) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if (t_m <= 1.45e-14) {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l_m)) / (1.0 / Math.cbrt((Math.sin(k) / l_m)))), 3.0) * t_3);
	} else if (t_m <= 5e+102) {
		tmp = ((2.0 / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) / (t_2 / l_m)) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * Math.pow(((t_m / (t_4 * t_4)) / (Math.cbrt(l_m) / Math.cbrt(Math.sin(k)))), 3.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	t_3 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	t_4 = cbrt(sqrt(l_m))
	tmp = 0.0
	if (t_m <= 7.8e-159)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif (t_m <= 1.45e-14)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l_m)) / Float64(1.0 / cbrt(Float64(sin(k) / l_m)))) ^ 3.0) * t_3));
	elseif (t_m <= 5e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) / Float64(t_2 / l_m)) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / Float64(t_3 * (Float64(Float64(t_m / Float64(t_4 * t_4)) / Float64(cbrt(l_m) / cbrt(sin(k)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sqrt[l$95$m], $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-159], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e-14], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+102], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[Power[N[(N[(t$95$m / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l$95$m, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.79999999999999953e-159

   1. Initial program 47.3%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. associate-*r* 62.1%

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

      3. times-frac 62.8%

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

   6. Simplified 62.8%

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

   if 7.79999999999999953e-159 < t < 1.4500000000000001e-14

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. *-commutative 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-/r* 69.4%

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

      5. distribute-rgt-in 69.4%

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

      6. unpow2 69.4%

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

      7. times-frac 69.5%

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

      8. sqr-neg 69.5%

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

      9. times-frac 69.4%

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

      10. unpow2 69.4%

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

      11. distribute-rgt-in 69.4%

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

      12. +-commutative 69.4%

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

   3. Simplified 69.4%

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

      1. associate-*l/ 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. add-cube-cbrt 72.7%

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

      2. pow3 72.7%

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

      3. associate-/l* 72.8%

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

      4. cbrt-div 72.7%

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

      5. cbrt-div 72.5%

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

      6. unpow3 72.5%

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

      7. add-cbrt-cube 92.3%

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

   7. Applied egg-rr 92.3%

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

      1. clear-num 92.3%

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

      2. cbrt-div 92.2%

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

      3. metadata-eval 92.2%

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

   9. Applied egg-rr 92.2%

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

   if 1.4500000000000001e-14 < t < 5e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. add-sqr-sqrt 84.1%

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

      3. times-frac 88.0%

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

      4. metadata-eval 88.0%

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

      5. associate-+r+ 88.0%

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

      6. add-sqr-sqrt 88.0%

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

      7. hypot-1-def 88.0%

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

      8. unpow2 88.0%

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

      9. hypot-1-def 88.0%

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

   5. Applied egg-rr 91.8%

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

      1. associate-/l* 91.8%

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

      2. associate-*r* 91.8%

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

   7. Simplified 91.8%

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

   if 5e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. associate-*l/ 61.4%

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

   5. Applied egg-rr 61.4%

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

      1. add-cube-cbrt 61.4%

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

      2. pow3 61.4%

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

      3. associate-/l* 61.1%

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

      4. cbrt-div 61.1%

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

      5. cbrt-div 61.1%

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

      6. unpow3 61.1%

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

      7. add-cbrt-cube 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. cbrt-div 91.3%

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

   9. Applied egg-rr 91.3%

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

       1. pow1/3 44.8%

          \[\leadsto \frac{2}{{\left(\frac{\frac{t}{\color{blue}{{\ell}^{0.3333333333333333}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

       2. add-sqr-sqrt 44.8%

          \[\leadsto \frac{2}{{\left(\frac{\frac{t}{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

       3. unpow-prod-down 44.8%

          \[\leadsto \frac{2}{{\left(\frac{\frac{t}{\color{blue}{{\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   11. Applied egg-rr 44.8%

       \[\leadsto \frac{2}{{\left(\frac{\frac{t}{\color{blue}{{\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. unpow1/3 44.9%

          \[\leadsto \frac{2}{{\left(\frac{\frac{t}{\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

       2. unpow1/3 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{1}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)}}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\ell}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3}}\\ \end{array} \]

Alternative 2: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t_m}\right)}^{2}\\ t_3 := \left(\tan k \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{l_m \cdot l_m}\right)\right) \cdot \left(1 + \left(1 + t_2\right)\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_3 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + t_2\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{l_m} \cdot \frac{t_m}{l_m}\right)\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{2}{{\left({t_m}^{1.5} \cdot \frac{k}{\frac{l_m}{\sqrt{2}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{l_m}{\sin k}} \cdot \sqrt{\frac{t_m}{\cos k}}\right)}^{-2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3
         (*
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l_m l_m))))
          (+ 1.0 (+ 1.0 t_2)))))
   (*
    t_s
    (if (<= t_3 5e+293)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 t_2))
        (* (sin k) (* (/ (pow t_m 2.0) l_m) (/ t_m l_m)))))
      (if (<= t_3 INFINITY)
        (/ 2.0 (pow (* (pow t_m 1.5) (/ k (/ l_m (sqrt 2.0)))) 2.0))
        (*
         2.0
         (pow (* (/ k (/ l_m (sin k))) (sqrt (/ t_m (cos k)))) -2.0)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2));
	double tmp;
	if (t_3 <= 5e+293) {
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * (sin(k) * ((pow(t_m, 2.0) / l_m) * (t_m / l_m))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = 2.0 / pow((pow(t_m, 1.5) * (k / (l_m / sqrt(2.0)))), 2.0);
	} else {
		tmp = 2.0 * pow(((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))), -2.0);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2));
	double tmp;
	if (t_3 <= 5e+293) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l_m) * (t_m / l_m))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / Math.pow((Math.pow(t_m, 1.5) * (k / (l_m / Math.sqrt(2.0)))), 2.0);
	} else {
		tmp = 2.0 * Math.pow(((k / (l_m / Math.sin(k))) * Math.sqrt((t_m / Math.cos(k)))), -2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow((k / t_m), 2.0)
	t_3 = (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2))
	tmp = 0
	if t_3 <= 5e+293:
		tmp = 2.0 / ((math.tan(k) * (2.0 + t_2)) * (math.sin(k) * ((math.pow(t_m, 2.0) / l_m) * (t_m / l_m))))
	elif t_3 <= math.inf:
		tmp = 2.0 / math.pow((math.pow(t_m, 1.5) * (k / (l_m / math.sqrt(2.0)))), 2.0)
	else:
		tmp = 2.0 * math.pow(((k / (l_m / math.sin(k))) * math.sqrt((t_m / math.cos(k)))), -2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l_m * l_m)))) * Float64(1.0 + Float64(1.0 + t_2)))
	tmp = 0.0
	if (t_3 <= 5e+293)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l_m) * Float64(t_m / l_m)))));
	elseif (t_3 <= Inf)
		tmp = Float64(2.0 / (Float64((t_m ^ 1.5) * Float64(k / Float64(l_m / sqrt(2.0)))) ^ 2.0));
	else
		tmp = Float64(2.0 * (Float64(Float64(k / Float64(l_m / sin(k))) * sqrt(Float64(t_m / cos(k)))) ^ -2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (k / t_m) ^ 2.0;
	t_3 = (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2));
	tmp = 0.0;
	if (t_3 <= 5e+293)
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * (sin(k) * (((t_m ^ 2.0) / l_m) * (t_m / l_m))));
	elseif (t_3 <= Inf)
		tmp = 2.0 / (((t_m ^ 1.5) * (k / (l_m / sqrt(2.0)))) ^ 2.0);
	else
		tmp = 2.0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ^ -2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 5e+293], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(2.0 / N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k / N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $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 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 5.00000000000000033e293

   1. Initial program 79.8%

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

      1. associate-*l* 79.8%

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

      2. *-commutative 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/r* 86.2%

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

      5. distribute-rgt-in 86.2%

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

      6. unpow2 86.2%

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

      7. times-frac 70.3%

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

      8. sqr-neg 70.3%

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

      9. times-frac 86.2%

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

      10. unpow2 86.2%

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

      11. distribute-rgt-in 86.2%

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

      12. +-commutative 86.2%

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

   3. Simplified 86.2%

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

      1. associate-/r* 79.8%

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

      2. unpow3 79.8%

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

      3. times-frac 89.3%

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

      4. pow2 89.3%

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

   5. Applied egg-rr 89.3%

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

   if 5.00000000000000033e293 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 76.9%

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

      1. associate-*l* 76.9%

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

      2. *-commutative 76.9%

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

      3. *-commutative 76.9%

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

      4. associate-/r* 76.9%

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

      5. distribute-rgt-in 76.9%

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

      6. unpow2 76.9%

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

      7. times-frac 61.8%

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

      8. sqr-neg 61.8%

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

      9. times-frac 76.9%

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

      10. unpow2 76.9%

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

      11. distribute-rgt-in 76.9%

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

      12. +-commutative 76.9%

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

   3. Simplified 76.9%

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

      1. add-sqr-sqrt 76.9%

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

      2. pow2 76.9%

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

   5. Applied egg-rr 84.4%

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

      1. associate-*r* 84.4%

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

   7. Simplified 84.4%

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

   8. Taylor expanded in k around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/l* 56.7%

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

   10. Simplified 56.7%

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

   11. Taylor expanded in k around 0 56.8%

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

       1. associate-*l/ 56.7%

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

       2. *-commutative 56.7%

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

       3. metadata-eval 56.7%

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

       4. pow-sqr 56.8%

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

       5. rem-sqrt-square 65.7%

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

       6. unpow1 65.7%

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

       7. sqr-pow 65.5%

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

       8. fabs-sqr 65.5%

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

       9. sqr-pow 65.7%

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

       10. unpow1 65.7%

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

       11. associate-/r/ 65.7%

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

   13. Simplified 65.7%

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

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

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/r* 7.1%

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

      5. distribute-rgt-in 7.1%

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

      6. unpow2 7.1%

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

      7. times-frac 5.8%

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

      8. sqr-neg 5.8%

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

      9. times-frac 7.1%

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

      10. unpow2 7.1%

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

      11. distribute-rgt-in 7.1%

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

      12. +-commutative 7.1%

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

   3. Simplified 7.1%

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

      1. add-sqr-sqrt 4.1%

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

      2. pow2 4.1%

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

   5. Applied egg-rr 9.4%

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

      1. associate-*r* 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 50.9%

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

      1. associate-*l/ 48.0%

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

   10. Simplified 48.0%

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

       1. expm1-log1p-u 47.3%

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

       2. expm1-udef 36.4%

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

       3. div-inv 36.4%

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

       4. pow-flip 36.4%

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

       5. associate-/l* 36.1%

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

       6. metadata-eval 36.1%

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

   12. Applied egg-rr 36.1%

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

       1. expm1-def 49.4%

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

       2. expm1-log1p 50.0%

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

       3. associate-/r/ 51.1%

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

       4. associate-/l* 51.2%

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

   14. Simplified 51.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{2}{{\left({t}^{1.5} \cdot \frac{k}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \end{array} \]

Alternative 3: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t_m}\right)\right)\\ t_3 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_4 := \frac{t_m}{\sqrt[3]{l_m}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_4}{\frac{1}{\sqrt[3]{\frac{\sin k}{l_m}}}}\right)}^{3} \cdot t_3}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)}}{\frac{t_2}{l_m}} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot {\left(\frac{t_4}{\frac{\sqrt[3]{l_m}}{\sqrt[3]{\sin k}}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k t_m))))
        (t_3 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))
        (t_4 (/ t_m (cbrt l_m))))
   (*
    t_s
    (if (<= t_m 5.9e-161)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (<= t_m 7.5e-14)
        (/ 2.0 (* (pow (/ t_4 (/ 1.0 (cbrt (/ (sin k) l_m)))) 3.0) t_3))
        (if (<= t_m 5.6e+102)
          (*
           (/ (/ 2.0 (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ t_2 l_m))
           (/ l_m t_2))
          (/ 2.0 (* t_3 (pow (/ t_4 (/ (cbrt l_m) (cbrt (sin k)))) 3.0)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double t_3 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double t_4 = t_m / cbrt(l_m);
	double tmp;
	if (t_m <= 5.9e-161) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if (t_m <= 7.5e-14) {
		tmp = 2.0 / (pow((t_4 / (1.0 / cbrt((sin(k) / l_m)))), 3.0) * t_3);
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (sin(k) * (tan(k) * pow(t_m, 3.0)))) / (t_2 / l_m)) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * pow((t_4 / (cbrt(l_m) / cbrt(sin(k)))), 3.0));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double t_3 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double t_4 = t_m / Math.cbrt(l_m);
	double tmp;
	if (t_m <= 5.9e-161) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if (t_m <= 7.5e-14) {
		tmp = 2.0 / (Math.pow((t_4 / (1.0 / Math.cbrt((Math.sin(k) / l_m)))), 3.0) * t_3);
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) / (t_2 / l_m)) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * Math.pow((t_4 / (Math.cbrt(l_m) / Math.cbrt(Math.sin(k)))), 3.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	t_3 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	t_4 = Float64(t_m / cbrt(l_m))
	tmp = 0.0
	if (t_m <= 5.9e-161)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif (t_m <= 7.5e-14)
		tmp = Float64(2.0 / Float64((Float64(t_4 / Float64(1.0 / cbrt(Float64(sin(k) / l_m)))) ^ 3.0) * t_3));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) / Float64(t_2 / l_m)) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / Float64(t_3 * (Float64(t_4 / Float64(cbrt(l_m) / cbrt(sin(k)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.9e-161], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-14], N[(2.0 / N[(N[Power[N[(t$95$4 / N[(1.0 / N[Power[N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[Power[N[(t$95$4 / N[(N[Power[l$95$m, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 5.9000000000000002e-161

   1. Initial program 47.3%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. associate-*r* 62.1%

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

      3. times-frac 62.8%

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

   6. Simplified 62.8%

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

   if 5.9000000000000002e-161 < t < 7.4999999999999996e-14

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. *-commutative 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-/r* 69.4%

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

      5. distribute-rgt-in 69.4%

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

      6. unpow2 69.4%

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

      7. times-frac 69.5%

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

      8. sqr-neg 69.5%

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

      9. times-frac 69.4%

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

      10. unpow2 69.4%

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

      11. distribute-rgt-in 69.4%

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

      12. +-commutative 69.4%

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

   3. Simplified 69.4%

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

      1. associate-*l/ 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. add-cube-cbrt 72.7%

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

      2. pow3 72.7%

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

      3. associate-/l* 72.8%

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

      4. cbrt-div 72.7%

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

      5. cbrt-div 72.5%

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

      6. unpow3 72.5%

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

      7. add-cbrt-cube 92.3%

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

   7. Applied egg-rr 92.3%

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

      1. clear-num 92.3%

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

      2. cbrt-div 92.2%

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

      3. metadata-eval 92.2%

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

   9. Applied egg-rr 92.2%

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

   if 7.4999999999999996e-14 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. add-sqr-sqrt 84.1%

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

      3. times-frac 88.0%

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

      4. metadata-eval 88.0%

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

      5. associate-+r+ 88.0%

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

      6. add-sqr-sqrt 88.0%

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

      7. hypot-1-def 88.0%

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

      8. unpow2 88.0%

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

      9. hypot-1-def 88.0%

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

   5. Applied egg-rr 91.8%

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

      1. associate-/l* 91.8%

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

      2. associate-*r* 91.8%

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

   7. Simplified 91.8%

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

   if 5.60000000000000037e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. associate-*l/ 61.4%

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

   5. Applied egg-rr 61.4%

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

      1. add-cube-cbrt 61.4%

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

      2. pow3 61.4%

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

      3. associate-/l* 61.1%

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

      4. cbrt-div 61.1%

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

      5. cbrt-div 61.1%

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

      6. unpow3 61.1%

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

      7. add-cbrt-cube 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. cbrt-div 91.3%

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

   9. Applied egg-rr 91.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{1}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)}}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\ell}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3}}\\ \end{array} \]

Alternative 4: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_3 := \tan k \cdot t_2\\ t_4 := \frac{t_m}{\sqrt[3]{l_m}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_4}{\frac{1}{\sqrt[3]{\frac{\sin k}{l_m}}}}\right)}^{3} \cdot t_3}\\ \mathbf{elif}\;t_m \leq 3.65 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot {\left(\frac{t_4}{\frac{\sqrt[3]{l_m}}{\sqrt[3]{\sin k}}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0)))
        (t_3 (* (tan k) t_2))
        (t_4 (/ t_m (cbrt l_m))))
   (*
    t_s
    (if (<= t_m 7.6e-159)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (<= t_m 1.55e-14)
        (/ 2.0 (* (pow (/ t_4 (/ 1.0 (cbrt (/ (sin k) l_m)))) 3.0) t_3))
        (if (<= t_m 3.65e+102)
          (* (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ l_m t_2))
          (/ 2.0 (* t_3 (pow (/ t_4 (/ (cbrt l_m) (cbrt (sin k)))) 3.0)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double t_3 = tan(k) * t_2;
	double t_4 = t_m / cbrt(l_m);
	double tmp;
	if (t_m <= 7.6e-159) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if (t_m <= 1.55e-14) {
		tmp = 2.0 / (pow((t_4 / (1.0 / cbrt((sin(k) / l_m)))), 3.0) * t_3);
	} else if (t_m <= 3.65e+102) {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * pow((t_4 / (cbrt(l_m) / cbrt(sin(k)))), 3.0));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double t_3 = Math.tan(k) * t_2;
	double t_4 = t_m / Math.cbrt(l_m);
	double tmp;
	if (t_m <= 7.6e-159) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if (t_m <= 1.55e-14) {
		tmp = 2.0 / (Math.pow((t_4 / (1.0 / Math.cbrt((Math.sin(k) / l_m)))), 3.0) * t_3);
	} else if (t_m <= 3.65e+102) {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * Math.pow((t_4 / (Math.cbrt(l_m) / Math.cbrt(Math.sin(k)))), 3.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	t_3 = Float64(tan(k) * t_2)
	t_4 = Float64(t_m / cbrt(l_m))
	tmp = 0.0
	if (t_m <= 7.6e-159)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif (t_m <= 1.55e-14)
		tmp = Float64(2.0 / Float64((Float64(t_4 / Float64(1.0 / cbrt(Float64(sin(k) / l_m)))) ^ 3.0) * t_3));
	elseif (t_m <= 3.65e+102)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / Float64(t_3 * (Float64(t_4 / Float64(cbrt(l_m) / cbrt(sin(k)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.6e-159], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.55e-14], N[(2.0 / N[(N[Power[N[(t$95$4 / N[(1.0 / N[Power[N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.65e+102], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[Power[N[(t$95$4 / N[(N[Power[l$95$m, 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.6000000000000002e-159

   1. Initial program 47.3%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. associate-*r* 62.1%

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

      3. times-frac 62.8%

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

   6. Simplified 62.8%

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

   if 7.6000000000000002e-159 < t < 1.55000000000000002e-14

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. *-commutative 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-/r* 69.4%

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

      5. distribute-rgt-in 69.4%

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

      6. unpow2 69.4%

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

      7. times-frac 69.5%

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

      8. sqr-neg 69.5%

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

      9. times-frac 69.4%

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

      10. unpow2 69.4%

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

      11. distribute-rgt-in 69.4%

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

      12. +-commutative 69.4%

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

   3. Simplified 69.4%

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

      1. associate-*l/ 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. add-cube-cbrt 72.7%

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

      2. pow3 72.7%

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

      3. associate-/l* 72.8%

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

      4. cbrt-div 72.7%

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

      5. cbrt-div 72.5%

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

      6. unpow3 72.5%

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

      7. add-cbrt-cube 92.3%

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

   7. Applied egg-rr 92.3%

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

      1. clear-num 92.3%

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

      2. cbrt-div 92.2%

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

      3. metadata-eval 92.2%

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

   9. Applied egg-rr 92.2%

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

   if 1.55000000000000002e-14 < t < 3.64999999999999995e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 3.64999999999999995e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. associate-*l/ 61.4%

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

   5. Applied egg-rr 61.4%

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

      1. add-cube-cbrt 61.4%

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

      2. pow3 61.4%

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

      3. associate-/l* 61.1%

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

      4. cbrt-div 61.1%

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

      5. cbrt-div 61.1%

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

      6. unpow3 61.1%

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

      7. add-cbrt-cube 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. cbrt-div 91.3%

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

   9. Applied egg-rr 91.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{1}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 3.65 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3}}\\ \end{array} \]

Alternative 5: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 1.06 \cdot 10^{-13} \lor \neg \left(t_m \leq 4.5 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t_m}{\sqrt[3]{l_m}}}{\frac{1}{\sqrt[3]{\frac{\sin k}{l_m}}}}\right)}^{3} \cdot \left(\tan k \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.1e-158)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (or (<= t_m 1.06e-13) (not (<= t_m 4.5e+102)))
        (/
         2.0
         (*
          (pow (/ (/ t_m (cbrt l_m)) (/ 1.0 (cbrt (/ (sin k) l_m)))) 3.0)
          (* (tan k) t_2)))
        (*
         (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0))))
         (/ l_m t_2)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.1e-158) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if ((t_m <= 1.06e-13) || !(t_m <= 4.5e+102)) {
		tmp = 2.0 / (pow(((t_m / cbrt(l_m)) / (1.0 / cbrt((sin(k) / l_m)))), 3.0) * (tan(k) * t_2));
	} else {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.1e-158) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if ((t_m <= 1.06e-13) || !(t_m <= 4.5e+102)) {
		tmp = 2.0 / (Math.pow(((t_m / Math.cbrt(l_m)) / (1.0 / Math.cbrt((Math.sin(k) / l_m)))), 3.0) * (Math.tan(k) * t_2));
	} else {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.1e-158)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif ((t_m <= 1.06e-13) || !(t_m <= 4.5e+102))
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / cbrt(l_m)) / Float64(1.0 / cbrt(Float64(sin(k) / l_m)))) ^ 3.0) * Float64(tan(k) * t_2)));
	else
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-158], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 1.06e-13], N[Not[LessEqual[t$95$m, 4.5e+102]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1000000000000001e-158

   1. Initial program 47.3%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. associate-*r* 62.1%

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

      3. times-frac 62.8%

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

   6. Simplified 62.8%

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

   if 1.1000000000000001e-158 < t < 1.06e-13 or 4.50000000000000021e102 < t

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. *-commutative 59.8%

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

      3. *-commutative 59.8%

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

      4. associate-/r* 65.0%

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

      5. distribute-rgt-in 65.0%

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

      6. unpow2 65.0%

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

      7. times-frac 58.7%

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

      8. sqr-neg 58.7%

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

      9. times-frac 65.0%

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

      10. unpow2 65.0%

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

      11. distribute-rgt-in 65.0%

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

      12. +-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. associate-*l/ 66.5%

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

   5. Applied egg-rr 66.5%

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

      1. add-cube-cbrt 66.5%

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

      2. pow3 66.5%

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

      3. associate-/l* 66.4%

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

      4. cbrt-div 66.3%

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

      5. cbrt-div 66.3%

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

      6. unpow3 66.3%

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

      7. add-cbrt-cube 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. clear-num 88.9%

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

      2. cbrt-div 88.9%

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

      3. metadata-eval 88.9%

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

   9. Applied egg-rr 88.9%

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

   if 1.06e-13 < t < 4.50000000000000021e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-13} \lor \neg \left(t \leq 4.5 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{1}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 6: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 5.8 \cdot 10^{-12} \lor \neg \left(t_m \leq 3.7 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_2\right) \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{l_m}}}{\sqrt[3]{\frac{l_m}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 7.8e-159)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (or (<= t_m 5.8e-12) (not (<= t_m 3.7e+102)))
        (/
         2.0
         (*
          (* (tan k) t_2)
          (pow (/ (/ t_m (cbrt l_m)) (cbrt (/ l_m (sin k)))) 3.0)))
        (*
         (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0))))
         (/ l_m t_2)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 7.8e-159) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if ((t_m <= 5.8e-12) || !(t_m <= 3.7e+102)) {
		tmp = 2.0 / ((tan(k) * t_2) * pow(((t_m / cbrt(l_m)) / cbrt((l_m / sin(k)))), 3.0));
	} else {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 7.8e-159) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if ((t_m <= 5.8e-12) || !(t_m <= 3.7e+102)) {
		tmp = 2.0 / ((Math.tan(k) * t_2) * Math.pow(((t_m / Math.cbrt(l_m)) / Math.cbrt((l_m / Math.sin(k)))), 3.0));
	} else {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.8e-159)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif ((t_m <= 5.8e-12) || !(t_m <= 3.7e+102))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_2) * (Float64(Float64(t_m / cbrt(l_m)) / cbrt(Float64(l_m / sin(k)))) ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-159], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 5.8e-12], N[Not[LessEqual[t$95$m, 3.7e+102]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.79999999999999953e-159

   1. Initial program 47.3%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. associate-*r* 62.1%

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

      3. times-frac 62.8%

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

   6. Simplified 62.8%

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

   if 7.79999999999999953e-159 < t < 5.8000000000000003e-12 or 3.70000000000000023e102 < t

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. *-commutative 59.8%

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

      3. *-commutative 59.8%

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

      4. associate-/r* 65.0%

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

      5. distribute-rgt-in 65.0%

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

      6. unpow2 65.0%

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

      7. times-frac 58.7%

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

      8. sqr-neg 58.7%

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

      9. times-frac 65.0%

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

      10. unpow2 65.0%

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

      11. distribute-rgt-in 65.0%

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

      12. +-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. associate-*l/ 66.5%

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

   5. Applied egg-rr 66.5%

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

      1. add-cube-cbrt 66.5%

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

      2. pow3 66.5%

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

      3. associate-/l* 66.4%

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

      4. cbrt-div 66.3%

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

      5. cbrt-div 66.3%

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

      6. unpow3 66.3%

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

      7. add-cbrt-cube 88.9%

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

   7. Applied egg-rr 88.9%

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

   if 5.8000000000000003e-12 < t < 3.70000000000000023e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-12} \lor \neg \left(t \leq 3.7 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 7: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_3 := \tan k \cdot t_2\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{t_3 \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{l_m}}{l_m}}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\sin k \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{l_m}}}{\sqrt[3]{l_m}}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))) (t_3 (* (tan k) t_2)))
   (*
    t_s
    (if (<= t_m 1.85e-87)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (<= t_m 8.6e-13)
        (/ 2.0 (* t_3 (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
        (if (<= t_m 5.6e+102)
          (* (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ l_m t_2))
          (/
           2.0
           (*
            t_3
            (* (sin k) (pow (/ (/ t_m (cbrt l_m)) (cbrt l_m)) 3.0))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double t_3 = tan(k) * t_2;
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if (t_m <= 8.6e-13) {
		tmp = 2.0 / (t_3 * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * (sin(k) * pow(((t_m / cbrt(l_m)) / cbrt(l_m)), 3.0)));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double t_3 = Math.tan(k) * t_2;
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if (t_m <= 8.6e-13) {
		tmp = 2.0 / (t_3 * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * (Math.sin(k) * Math.pow(((t_m / Math.cbrt(l_m)) / Math.cbrt(l_m)), 3.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	t_3 = Float64(tan(k) * t_2)
	tmp = 0.0
	if (t_m <= 1.85e-87)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif (t_m <= 8.6e-13)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(sin(k) * (Float64(Float64(t_m / cbrt(l_m)) / cbrt(l_m)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.85e-87], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e-13], N[(2.0 / N[(t$95$3 * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.8500000000000001e-87

   1. Initial program 48.4%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in k around inf 64.1%

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

      1. associate-*r/ 64.1%

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

      2. associate-*r* 64.1%

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

      3. times-frac 64.7%

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

   6. Simplified 64.7%

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

   if 1.8500000000000001e-87 < t < 8.5999999999999997e-13

   1. Initial program 79.1%

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

      1. associate-*l* 79.0%

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

      2. *-commutative 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/r* 79.0%

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

      5. distribute-rgt-in 79.0%

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

      6. unpow2 79.0%

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

      7. times-frac 79.2%

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

      8. sqr-neg 79.2%

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

      9. times-frac 79.0%

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

      10. unpow2 79.0%

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

      11. distribute-rgt-in 79.0%

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

      12. +-commutative 79.0%

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

   3. Simplified 79.0%

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

      1. associate-*l/ 85.9%

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

   5. Applied egg-rr 85.9%

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

   if 8.5999999999999997e-13 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 5.60000000000000037e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. add-cube-cbrt 61.4%

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

      2. *-un-lft-identity 61.4%

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

      3. times-frac 61.4%

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

      4. pow2 61.4%

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

      5. cbrt-div 61.4%

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

      6. rem-cbrt-cube 61.4%

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

      7. cbrt-div 61.4%

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

      8. rem-cbrt-cube 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. add-cube-cbrt 83.2%

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

      2. pow3 83.2%

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

      3. frac-times 72.5%

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

      4. unpow2 72.5%

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

      5. *-un-lft-identity 72.5%

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

      6. cbrt-div 72.5%

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

      7. add-cbrt-cube 83.2%

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 8: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_3 := \tan k \cdot t_2\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{t_m \cdot {k}^{2}} \cdot \frac{{l_m}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t_m \leq 7.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t_3 \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{l_m}}{l_m}}\\ \mathbf{elif}\;t_m \leq 2.12 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{l_m}} \cdot \sqrt[3]{\sin k}\right)}^{3}}{l_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))) (t_3 (* (tan k) t_2)))
   (*
    t_s
    (if (<= t_m 4.6e-87)
      (*
       (/ 2.0 (* t_m (pow k 2.0)))
       (/ (* (pow l_m 2.0) (cos k)) (pow (sin k) 2.0)))
      (if (<= t_m 7.6e-12)
        (/ 2.0 (* t_3 (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
        (if (<= t_m 2.12e+102)
          (* (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ l_m t_2))
          (/
           2.0
           (*
            t_3
            (/ (pow (* (/ t_m (cbrt l_m)) (cbrt (sin k))) 3.0) l_m)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double t_3 = tan(k) * t_2;
	double tmp;
	if (t_m <= 4.6e-87) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l_m, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if (t_m <= 7.6e-12) {
		tmp = 2.0 / (t_3 * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else if (t_m <= 2.12e+102) {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * (pow(((t_m / cbrt(l_m)) * cbrt(sin(k))), 3.0) / l_m));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double t_3 = Math.tan(k) * t_2;
	double tmp;
	if (t_m <= 4.6e-87) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l_m, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if (t_m <= 7.6e-12) {
		tmp = 2.0 / (t_3 * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else if (t_m <= 2.12e+102) {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / (t_3 * (Math.pow(((t_m / Math.cbrt(l_m)) * Math.cbrt(Math.sin(k))), 3.0) / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	t_3 = Float64(tan(k) * t_2)
	tmp = 0.0
	if (t_m <= 4.6e-87)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l_m ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif (t_m <= 7.6e-12)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	elseif (t_m <= 2.12e+102)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64((Float64(Float64(t_m / cbrt(l_m)) * cbrt(sin(k))) ^ 3.0) / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.6e-87], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.6e-12], N[(2.0 / N[(t$95$3 * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.12e+102], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[Power[N[(N[(t$95$m / N[Power[l$95$m, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.6000000000000003e-87

   1. Initial program 48.4%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in k around inf 64.1%

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

      1. associate-*r/ 64.1%

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

      2. associate-*r* 64.1%

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

      3. times-frac 64.7%

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

   6. Simplified 64.7%

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

   if 4.6000000000000003e-87 < t < 7.59999999999999993e-12

   1. Initial program 79.1%

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

      1. associate-*l* 79.0%

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

      2. *-commutative 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/r* 79.0%

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

      5. distribute-rgt-in 79.0%

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

      6. unpow2 79.0%

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

      7. times-frac 79.2%

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

      8. sqr-neg 79.2%

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

      9. times-frac 79.0%

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

      10. unpow2 79.0%

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

      11. distribute-rgt-in 79.0%

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

      12. +-commutative 79.0%

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

   3. Simplified 79.0%

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

      1. associate-*l/ 85.9%

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

   5. Applied egg-rr 85.9%

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

   if 7.59999999999999993e-12 < t < 2.12000000000000003e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 2.12000000000000003e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. associate-*l/ 61.4%

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

   5. Applied egg-rr 61.4%

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

      1. add-cube-cbrt 61.4%

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

      2. pow3 61.4%

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

      3. cbrt-prod 61.4%

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

      4. cbrt-div 61.4%

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

      5. unpow3 61.4%

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

      6. add-cbrt-cube 83.3%

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

   7. Applied egg-rr 83.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 2.12 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right)}^{3}}{\ell}}\\ \end{array} \]

Alternative 9: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{l_m}{\sin k}} \cdot \sqrt{\frac{t_m}{\cos k}}\right)}^{-2}\\ \mathbf{elif}\;t_m \leq 2.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_2\right) \cdot \frac{\frac{k}{\frac{l_m}{{t_m}^{3}}}}{l_m}}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot {t_m}^{1.5}}{\frac{l_m}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 3.8e-109)
      (* 2.0 (pow (* (/ k (/ l_m (sin k))) (sqrt (/ t_m (cos k)))) -2.0))
      (if (<= t_m 2.25e-13)
        (/ 2.0 (* (* (tan k) t_2) (/ (/ k (/ l_m (pow t_m 3.0))) l_m)))
        (if (<= t_m 5.6e+102)
          (* (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ l_m t_2))
          (/ 2.0 (pow (/ (* k (pow t_m 1.5)) (/ l_m (sqrt 2.0))) 2.0))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.8e-109) {
		tmp = 2.0 * pow(((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))), -2.0);
	} else if (t_m <= 2.25e-13) {
		tmp = 2.0 / ((tan(k) * t_2) * ((k / (l_m / pow(t_m, 3.0))) / l_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / pow(((k * pow(t_m, 1.5)) / (l_m / sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 + ((k / t_m) ** 2.0d0)
    if (t_m <= 3.8d-109) then
        tmp = 2.0d0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ** (-2.0d0))
    else if (t_m <= 2.25d-13) then
        tmp = 2.0d0 / ((tan(k) * t_2) * ((k / (l_m / (t_m ** 3.0d0))) / l_m))
    else if (t_m <= 5.6d+102) then
        tmp = ((2.0d0 * l_m) / (sin(k) * (tan(k) * (t_m ** 3.0d0)))) * (l_m / t_2)
    else
        tmp = 2.0d0 / (((k * (t_m ** 1.5d0)) / (l_m / sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.8e-109) {
		tmp = 2.0 * Math.pow(((k / (l_m / Math.sin(k))) * Math.sqrt((t_m / Math.cos(k)))), -2.0);
	} else if (t_m <= 2.25e-13) {
		tmp = 2.0 / ((Math.tan(k) * t_2) * ((k / (l_m / Math.pow(t_m, 3.0))) / l_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / Math.pow(((k * Math.pow(t_m, 1.5)) / (l_m / Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = 2.0 + math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 3.8e-109:
		tmp = 2.0 * math.pow(((k / (l_m / math.sin(k))) * math.sqrt((t_m / math.cos(k)))), -2.0)
	elif t_m <= 2.25e-13:
		tmp = 2.0 / ((math.tan(k) * t_2) * ((k / (l_m / math.pow(t_m, 3.0))) / l_m))
	elif t_m <= 5.6e+102:
		tmp = ((2.0 * l_m) / (math.sin(k) * (math.tan(k) * math.pow(t_m, 3.0)))) * (l_m / t_2)
	else:
		tmp = 2.0 / math.pow(((k * math.pow(t_m, 1.5)) / (l_m / math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.8e-109)
		tmp = Float64(2.0 * (Float64(Float64(k / Float64(l_m / sin(k))) * sqrt(Float64(t_m / cos(k)))) ^ -2.0));
	elseif (t_m <= 2.25e-13)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_2) * Float64(Float64(k / Float64(l_m / (t_m ^ 3.0))) / l_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * (t_m ^ 1.5)) / Float64(l_m / sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = 2.0 + ((k / t_m) ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.8e-109)
		tmp = 2.0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ^ -2.0);
	elseif (t_m <= 2.25e-13)
		tmp = 2.0 / ((tan(k) * t_2) * ((k / (l_m / (t_m ^ 3.0))) / l_m));
	elseif (t_m <= 5.6e+102)
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * (t_m ^ 3.0)))) * (l_m / t_2);
	else
		tmp = 2.0 / (((k * (t_m ^ 1.5)) / (l_m / sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.8e-109], N[(2.0 * N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e-13], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(k / N[(l$95$m / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.80000000000000002e-109

   1. Initial program 47.5%

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

      1. associate-*l* 47.5%

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

      2. *-commutative 47.5%

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

      3. *-commutative 47.5%

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

      4. associate-/r* 53.2%

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

      5. distribute-rgt-in 53.2%

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

      6. unpow2 53.2%

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

      7. times-frac 39.9%

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

      8. sqr-neg 39.9%

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

      9. times-frac 53.2%

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

      10. unpow2 53.2%

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

      11. distribute-rgt-in 53.2%

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

      12. +-commutative 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 18.6%

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

      2. pow2 18.6%

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

   5. Applied egg-rr 21.5%

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

      1. associate-*r* 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 37.8%

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

      1. associate-*l/ 37.4%

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

   10. Simplified 37.4%

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

       1. expm1-log1p-u 37.0%

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

       2. expm1-udef 30.9%

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

       3. div-inv 30.9%

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

       4. pow-flip 30.9%

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

       5. associate-/l* 30.8%

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

       6. metadata-eval 30.8%

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

   12. Applied egg-rr 30.8%

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

       1. expm1-def 37.0%

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

       2. expm1-log1p 37.4%

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

       3. associate-/r/ 37.9%

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

       4. associate-/l* 37.9%

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

   14. Simplified 37.9%

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

   if 3.80000000000000002e-109 < t < 2.25e-13

   1. Initial program 82.8%

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

      1. associate-*l* 82.7%

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

      2. *-commutative 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.9%

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

      8. sqr-neg 82.9%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

      12. +-commutative 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 88.4%

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

   5. Applied egg-rr 88.4%

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

   6. Taylor expanded in k around 0 77.5%

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

      1. associate-/l* 82.9%

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

   8. Simplified 82.9%

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

   if 2.25e-13 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 5.60000000000000037e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. add-sqr-sqrt 49.8%

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

      2. pow2 49.8%

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

   5. Applied egg-rr 49.8%

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

      1. associate-*r* 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-/l* 61.1%

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

   10. Simplified 61.1%

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

       1. associate-*r/ 61.4%

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

       2. sqrt-pow1 86.3%

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

       3. metadata-eval 86.3%

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

   12. Applied egg-rr 86.3%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{t}^{1.5} \cdot k}{\frac{\ell}{\sqrt{2}}}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\frac{k}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot {t}^{1.5}}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \]

Alternative 10: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{l_m}{\sin k}} \cdot \sqrt{\frac{t_m}{\cos k}}\right)}^{-2}\\ \mathbf{elif}\;t_m \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_2\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{l_m}}{l_m}}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot l_m}{\sin k \cdot \left(\tan k \cdot {t_m}^{3}\right)} \cdot \frac{l_m}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot {t_m}^{1.5}}{\frac{l_m}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 3.8e-109)
      (* 2.0 (pow (* (/ k (/ l_m (sin k))) (sqrt (/ t_m (cos k)))) -2.0))
      (if (<= t_m 2.2e-13)
        (/ 2.0 (* (* (tan k) t_2) (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
        (if (<= t_m 5.6e+102)
          (* (/ (* 2.0 l_m) (* (sin k) (* (tan k) (pow t_m 3.0)))) (/ l_m t_2))
          (/ 2.0 (pow (/ (* k (pow t_m 1.5)) (/ l_m (sqrt 2.0))) 2.0))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.8e-109) {
		tmp = 2.0 * pow(((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))), -2.0);
	} else if (t_m <= 2.2e-13) {
		tmp = 2.0 / ((tan(k) * t_2) * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / pow(((k * pow(t_m, 1.5)) / (l_m / sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 + ((k / t_m) ** 2.0d0)
    if (t_m <= 3.8d-109) then
        tmp = 2.0d0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ** (-2.0d0))
    else if (t_m <= 2.2d-13) then
        tmp = 2.0d0 / ((tan(k) * t_2) * ((sin(k) * ((t_m ** 3.0d0) / l_m)) / l_m))
    else if (t_m <= 5.6d+102) then
        tmp = ((2.0d0 * l_m) / (sin(k) * (tan(k) * (t_m ** 3.0d0)))) * (l_m / t_2)
    else
        tmp = 2.0d0 / (((k * (t_m ** 1.5d0)) / (l_m / sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.8e-109) {
		tmp = 2.0 * Math.pow(((k / (l_m / Math.sin(k))) * Math.sqrt((t_m / Math.cos(k)))), -2.0);
	} else if (t_m <= 2.2e-13) {
		tmp = 2.0 / ((Math.tan(k) * t_2) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l_m) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l_m / t_2);
	} else {
		tmp = 2.0 / Math.pow(((k * Math.pow(t_m, 1.5)) / (l_m / Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = 2.0 + math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 3.8e-109:
		tmp = 2.0 * math.pow(((k / (l_m / math.sin(k))) * math.sqrt((t_m / math.cos(k)))), -2.0)
	elif t_m <= 2.2e-13:
		tmp = 2.0 / ((math.tan(k) * t_2) * ((math.sin(k) * (math.pow(t_m, 3.0) / l_m)) / l_m))
	elif t_m <= 5.6e+102:
		tmp = ((2.0 * l_m) / (math.sin(k) * (math.tan(k) * math.pow(t_m, 3.0)))) * (l_m / t_2)
	else:
		tmp = 2.0 / math.pow(((k * math.pow(t_m, 1.5)) / (l_m / math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.8e-109)
		tmp = Float64(2.0 * (Float64(Float64(k / Float64(l_m / sin(k))) * sqrt(Float64(t_m / cos(k)))) ^ -2.0));
	elseif (t_m <= 2.2e-13)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_2) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l_m / t_2));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * (t_m ^ 1.5)) / Float64(l_m / sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = 2.0 + ((k / t_m) ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.8e-109)
		tmp = 2.0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ^ -2.0);
	elseif (t_m <= 2.2e-13)
		tmp = 2.0 / ((tan(k) * t_2) * ((sin(k) * ((t_m ^ 3.0) / l_m)) / l_m));
	elseif (t_m <= 5.6e+102)
		tmp = ((2.0 * l_m) / (sin(k) * (tan(k) * (t_m ^ 3.0)))) * (l_m / t_2);
	else
		tmp = 2.0 / (((k * (t_m ^ 1.5)) / (l_m / sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.8e-109], N[(2.0 * N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e-13], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.80000000000000002e-109

   1. Initial program 47.5%

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

      1. associate-*l* 47.5%

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

      2. *-commutative 47.5%

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

      3. *-commutative 47.5%

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

      4. associate-/r* 53.2%

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

      5. distribute-rgt-in 53.2%

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

      6. unpow2 53.2%

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

      7. times-frac 39.9%

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

      8. sqr-neg 39.9%

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

      9. times-frac 53.2%

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

      10. unpow2 53.2%

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

      11. distribute-rgt-in 53.2%

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

      12. +-commutative 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 18.6%

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

      2. pow2 18.6%

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

   5. Applied egg-rr 21.5%

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

      1. associate-*r* 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 37.8%

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

      1. associate-*l/ 37.4%

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

   10. Simplified 37.4%

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

       1. expm1-log1p-u 37.0%

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

       2. expm1-udef 30.9%

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

       3. div-inv 30.9%

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

       4. pow-flip 30.9%

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

       5. associate-/l* 30.8%

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

       6. metadata-eval 30.8%

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

   12. Applied egg-rr 30.8%

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

       1. expm1-def 37.0%

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

       2. expm1-log1p 37.4%

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

       3. associate-/r/ 37.9%

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

       4. associate-/l* 37.9%

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

   14. Simplified 37.9%

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

   if 3.80000000000000002e-109 < t < 2.19999999999999997e-13

   1. Initial program 82.8%

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

      1. associate-*l* 82.7%

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

      2. *-commutative 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.9%

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

      8. sqr-neg 82.9%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

      12. +-commutative 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 88.4%

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

   5. Applied egg-rr 88.4%

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

   if 2.19999999999999997e-13 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

      1. associate-*r* 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. times-frac 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. /-rgt-identity 87.9%

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

      2. associate-*l/ 89.7%

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

      3. associate-*r* 89.7%

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

   7. Simplified 89.7%

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

   if 5.60000000000000037e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. add-sqr-sqrt 49.8%

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

      2. pow2 49.8%

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

   5. Applied egg-rr 49.8%

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

      1. associate-*r* 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-/l* 61.1%

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

   10. Simplified 61.1%

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

       1. associate-*r/ 61.4%

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

       2. sqrt-pow1 86.3%

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

       3. metadata-eval 86.3%

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

   12. Applied egg-rr 86.3%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{t}^{1.5} \cdot k}{\frac{\ell}{\sqrt{2}}}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot {t}^{1.5}}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \]

Alternative 11: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{l_m}{\sin k}} \cdot \sqrt{\frac{t_m}{\cos k}}\right)}^{-2}\\ \mathbf{elif}\;t_m \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\frac{{t_m}^{3}}{l_m}}{l_m}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k}{t_m \cdot \frac{t_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot {t_m}^{1.5}}{\frac{l_m}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-109)
    (* 2.0 (pow (* (/ k (/ l_m (sin k))) (sqrt (/ t_m (cos k)))) -2.0))
    (if (<= t_m 4.2e+102)
      (/
       2.0
       (*
        (* (sin k) (/ (/ (pow t_m 3.0) l_m) l_m))
        (* (tan k) (+ 2.0 (/ k (* t_m (/ t_m k)))))))
      (/ 2.0 (pow (/ (* k (pow t_m 1.5)) (/ l_m (sqrt 2.0))) 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 3.8e-109) {
		tmp = 2.0 * pow(((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))), -2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = 2.0 / ((sin(k) * ((pow(t_m, 3.0) / l_m) / l_m)) * (tan(k) * (2.0 + (k / (t_m * (t_m / k))))));
	} else {
		tmp = 2.0 / pow(((k * pow(t_m, 1.5)) / (l_m / sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.8d-109) then
        tmp = 2.0d0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ** (-2.0d0))
    else if (t_m <= 4.2d+102) then
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 3.0d0) / l_m) / l_m)) * (tan(k) * (2.0d0 + (k / (t_m * (t_m / k))))))
    else
        tmp = 2.0d0 / (((k * (t_m ** 1.5d0)) / (l_m / sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 3.8e-109) {
		tmp = 2.0 * Math.pow(((k / (l_m / Math.sin(k))) * Math.sqrt((t_m / Math.cos(k)))), -2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t_m, 3.0) / l_m) / l_m)) * (Math.tan(k) * (2.0 + (k / (t_m * (t_m / k))))));
	} else {
		tmp = 2.0 / Math.pow(((k * Math.pow(t_m, 1.5)) / (l_m / Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 3.8e-109:
		tmp = 2.0 * math.pow(((k / (l_m / math.sin(k))) * math.sqrt((t_m / math.cos(k)))), -2.0)
	elif t_m <= 4.2e+102:
		tmp = 2.0 / ((math.sin(k) * ((math.pow(t_m, 3.0) / l_m) / l_m)) * (math.tan(k) * (2.0 + (k / (t_m * (t_m / k))))))
	else:
		tmp = 2.0 / math.pow(((k * math.pow(t_m, 1.5)) / (l_m / math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 3.8e-109)
		tmp = Float64(2.0 * (Float64(Float64(k / Float64(l_m / sin(k))) * sqrt(Float64(t_m / cos(k)))) ^ -2.0));
	elseif (t_m <= 4.2e+102)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l_m) / l_m)) * Float64(tan(k) * Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * (t_m ^ 1.5)) / Float64(l_m / sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 3.8e-109)
		tmp = 2.0 * (((k / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ^ -2.0);
	elseif (t_m <= 4.2e+102)
		tmp = 2.0 / ((sin(k) * (((t_m ^ 3.0) / l_m) / l_m)) * (tan(k) * (2.0 + (k / (t_m * (t_m / k))))));
	else
		tmp = 2.0 / (((k * (t_m ^ 1.5)) / (l_m / sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-109], N[(2.0 * N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e+102], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.80000000000000002e-109

   1. Initial program 47.5%

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

      1. associate-*l* 47.5%

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

      2. *-commutative 47.5%

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

      3. *-commutative 47.5%

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

      4. associate-/r* 53.2%

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

      5. distribute-rgt-in 53.2%

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

      6. unpow2 53.2%

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

      7. times-frac 39.9%

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

      8. sqr-neg 39.9%

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

      9. times-frac 53.2%

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

      10. unpow2 53.2%

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

      11. distribute-rgt-in 53.2%

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

      12. +-commutative 53.2%

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

   3. Simplified 53.2%

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

      1. add-sqr-sqrt 18.6%

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

      2. pow2 18.6%

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

   5. Applied egg-rr 21.5%

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

      1. associate-*r* 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in k around inf 37.8%

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

      1. associate-*l/ 37.4%

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

   10. Simplified 37.4%

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

       1. expm1-log1p-u 37.0%

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

       2. expm1-udef 30.9%

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

       3. div-inv 30.9%

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

       4. pow-flip 30.9%

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

       5. associate-/l* 30.8%

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

       6. metadata-eval 30.8%

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

   12. Applied egg-rr 30.8%

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

       1. expm1-def 37.0%

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

       2. expm1-log1p 37.4%

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

       3. associate-/r/ 37.9%

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

       4. associate-/l* 37.9%

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

   14. Simplified 37.9%

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

   if 3.80000000000000002e-109 < t < 4.20000000000000003e102

   1. Initial program 77.4%

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

      1. associate-*l* 77.4%

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

      2. *-commutative 77.4%

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

      3. *-commutative 77.4%

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

      4. associate-/r* 77.4%

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

      5. distribute-rgt-in 77.4%

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

      6. unpow2 77.4%

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

      7. times-frac 77.4%

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

      8. sqr-neg 77.4%

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

      9. times-frac 77.4%

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

      10. unpow2 77.4%

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

      11. distribute-rgt-in 77.4%

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

      12. +-commutative 77.4%

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

   3. Simplified 77.4%

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

      1. unpow2 77.4%

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

      2. clear-num 77.4%

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

      3. frac-times 77.5%

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

      4. *-un-lft-identity 77.5%

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

   5. Applied egg-rr 77.5%

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

   if 4.20000000000000003e102 < t

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 49.8%

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

      8. sqr-neg 49.8%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

      12. +-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. add-sqr-sqrt 49.8%

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

      2. pow2 49.8%

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

   5. Applied egg-rr 49.8%

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

      1. associate-*r* 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-/l* 61.1%

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

   10. Simplified 61.1%

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

       1. associate-*r/ 61.4%

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

       2. sqrt-pow1 86.3%

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

       3. metadata-eval 86.3%

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

   12. Applied egg-rr 86.3%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{-2}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot {t}^{1.5}}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \]

Alternative 12: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{l_m} \cdot \sqrt{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t_m}^{1.5} \cdot \frac{k}{\frac{l_m}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e-79)
    (/ 2.0 (pow (* (/ (pow k 2.0) l_m) (sqrt t_m)) 2.0))
    (/ 2.0 (pow (* (pow t_m 1.5) (/ k (/ l_m (sqrt 2.0)))) 2.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.5e-79) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l_m) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / pow((pow(t_m, 1.5) * (k / (l_m / sqrt(2.0)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.5d-79) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l_m) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 / (((t_m ** 1.5d0) * (k / (l_m / sqrt(2.0d0)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.5e-79) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l_m) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / Math.pow((Math.pow(t_m, 1.5) * (k / (l_m / Math.sqrt(2.0)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.5e-79:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l_m) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 / math.pow((math.pow(t_m, 1.5) * (k / (l_m / math.sqrt(2.0)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.5e-79)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l_m) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64((t_m ^ 1.5) * Float64(k / Float64(l_m / sqrt(2.0)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.5e-79)
		tmp = 2.0 / ((((k ^ 2.0) / l_m) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 / (((t_m ^ 1.5) * (k / (l_m / sqrt(2.0)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-79], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(k / N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999969e-79

   1. Initial program 48.9%

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

      1. associate-*l* 48.9%

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

      2. *-commutative 48.9%

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

      3. *-commutative 48.9%

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

      4. associate-/r* 54.5%

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

      5. distribute-rgt-in 54.5%

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

      6. unpow2 54.5%

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

      7. times-frac 41.5%

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

      8. sqr-neg 41.5%

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

      9. times-frac 54.5%

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

      10. unpow2 54.5%

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

      11. distribute-rgt-in 54.5%

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

      12. +-commutative 54.5%

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

   3. Simplified 54.5%

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

      1. add-sqr-sqrt 20.2%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      2. pow2 20.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 23.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* 23.1%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}} \]

   7. Simplified 23.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   8. Taylor expanded in k around inf 39.0%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. associate-*l/ 38.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   10. Simplified 38.6%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   11. Taylor expanded in k around 0 18.5%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]

   if 7.49999999999999969e-79 < t

   1. Initial program 63.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 63.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 63.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 68.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 68.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 68.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 62.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 62.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 68.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 68.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 68.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 68.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 51.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      2. pow2 51.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 55.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* 55.6%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}} \]

   7. Simplified 55.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   8. Taylor expanded in k around 0 71.3%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{t}^{3}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}}^{2}} \]

      2. associate-/l* 71.3%

         \[\leadsto \frac{2}{{\left(\sqrt{{t}^{3}} \cdot \color{blue}{\frac{k}{\frac{\ell}{\sqrt{2}}}}\right)}^{2}} \]

   10. Simplified 71.3%

       \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{t}^{3}} \cdot \frac{k}{\frac{\ell}{\sqrt{2}}}\right)}}^{2}} \]

   11. Taylor expanded in k around 0 71.3%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}}^{2}} \]
   12. Step-by-step derivation

       1. associate-*l/ 71.3%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{2}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \]

       2. *-commutative 71.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{t}^{3}} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}}^{2}} \]

       3. metadata-eval 71.3%

          \[\leadsto \frac{2}{{\left(\sqrt{{t}^{\color{blue}{\left(2 \cdot 1.5\right)}}} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       4. pow-sqr 71.3%

          \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       5. rem-sqrt-square 83.4%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left|{t}^{1.5}\right|} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       6. unpow1 83.4%

          \[\leadsto \frac{2}{{\left(\left|\color{blue}{{\left({t}^{1.5}\right)}^{1}}\right| \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       7. sqr-pow 83.3%

          \[\leadsto \frac{2}{{\left(\left|\color{blue}{{\left({t}^{1.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({t}^{1.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       8. fabs-sqr 83.3%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left({\left({t}^{1.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({t}^{1.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       9. sqr-pow 83.4%

          \[\leadsto \frac{2}{{\left(\color{blue}{{\left({t}^{1.5}\right)}^{1}} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       10. unpow1 83.4%

           \[\leadsto \frac{2}{{\left(\color{blue}{{t}^{1.5}} \cdot \left(\frac{k}{\ell} \cdot \sqrt{2}\right)\right)}^{2}} \]

       11. associate-/r/ 83.4%

           \[\leadsto \frac{2}{{\left({t}^{1.5} \cdot \color{blue}{\frac{k}{\frac{\ell}{\sqrt{2}}}}\right)}^{2}} \]

   13. Simplified 83.4%

       \[\leadsto \frac{2}{{\color{blue}{\left({t}^{1.5} \cdot \frac{k}{\frac{\ell}{\sqrt{2}}}\right)}}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t}^{1.5} \cdot \frac{k}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \]

Alternative 13: 72.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{l_m} \cdot \sqrt{t_m}\right)}^{2}}\\ \mathbf{elif}\;t_m \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{{t_m}^{3} \cdot \frac{k}{l_m}}{\frac{l_m}{2 \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{{\left(t_m \cdot \sqrt[3]{\frac{k}{l_m}}\right)}^{3}}{l_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-87)
    (/ 2.0 (pow (* (/ (pow k 2.0) l_m) (sqrt t_m)) 2.0))
    (if (<= t_m 3.3e+96)
      (/ 2.0 (/ (* (pow t_m 3.0) (/ k l_m)) (/ l_m (* 2.0 k))))
      (/ 2.0 (* (* 2.0 k) (/ (pow (* t_m (cbrt (/ k l_m))) 3.0) l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 5e-87) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l_m) * sqrt(t_m)), 2.0);
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (pow((t_m * cbrt((k / l_m))), 3.0) / l_m));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 5e-87) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l_m) * Math.sqrt(t_m)), 2.0);
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.pow((t_m * Math.cbrt((k / l_m))), 3.0) / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 5e-87)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l_m) * sqrt(t_m)) ^ 2.0));
	elseif (t_m <= 3.3e+96)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(k / l_m)) / Float64(l_m / Float64(2.0 * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64((Float64(t_m * cbrt(Float64(k / l_m))) ^ 3.0) / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-87], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+96], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Power[N[(k / l$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.00000000000000042e-87

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 54.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      2. pow2 19.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* 22.8%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}} \]

   7. Simplified 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   8. Taylor expanded in k around inf 38.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. associate-*l/ 38.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   10. Simplified 38.5%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   11. Taylor expanded in k around 0 18.1%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]

   if 5.00000000000000042e-87 < t < 3.29999999999999984e96

   1. Initial program 74.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 74.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 74.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 74.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 76.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 76.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 61.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 61.3%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 61.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 66.4%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 66.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 68.7%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{{t}^{3}}} \cdot \left(k \cdot 2\right)}{\ell}}} \]

       2. associate-/r/ 76.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}{\ell}} \]

   13. Applied egg-rr 76.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-/l* 78.6%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\ell} \cdot {t}^{3}}{\frac{\ell}{k \cdot 2}}}} \]

       2. *-commutative 78.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot 2}}} \]

   15. Simplified 78.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot 2}}}} \]

   if 3.29999999999999984e96 < t

   1. Initial program 54.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 52.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 52.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 63.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 63.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 63.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 63.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 63.5%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}} \cdot \sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}}\right) \cdot \sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       2. pow3 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}}\right)}^{3}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       3. associate-/r/ 63.2%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\color{blue}{\frac{k}{\ell} \cdot {t}^{3}}}\right)}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

       4. cbrt-prod 63.2%

          \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{{t}^{3}}\right)}}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

       5. unpow3 63.2%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}\right)}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

       6. add-cbrt-cube 82.0%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \color{blue}{t}\right)}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

   13. Applied egg-rr 82.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot t\right)}^{3}}}{\ell} \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{2 \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{{\left(t \cdot \sqrt[3]{\frac{k}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]

Alternative 14: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{t_m}{\frac{{l_m}^{2}}{{k}^{4}}}}\\ \mathbf{elif}\;t_m \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{{t_m}^{3} \cdot \frac{k}{l_m}}{\frac{l_m}{2 \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\frac{k}{{\left(\frac{\sqrt[3]{l_m}}{t_m}\right)}^{3}}}{l_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-88)
    (/ 2.0 (/ t_m (/ (pow l_m 2.0) (pow k 4.0))))
    (if (<= t_m 3.3e+96)
      (/ 2.0 (/ (* (pow t_m 3.0) (/ k l_m)) (/ l_m (* 2.0 k))))
      (/ 2.0 (* (* 2.0 k) (/ (/ k (pow (/ (cbrt l_m) t_m) 3.0)) l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.2e-88) {
		tmp = 2.0 / (t_m / (pow(l_m, 2.0) / pow(k, 4.0)));
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k / pow((cbrt(l_m) / t_m), 3.0)) / l_m));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.2e-88) {
		tmp = 2.0 / (t_m / (Math.pow(l_m, 2.0) / Math.pow(k, 4.0)));
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k / Math.pow((Math.cbrt(l_m) / t_m), 3.0)) / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 6.2e-88)
		tmp = Float64(2.0 / Float64(t_m / Float64((l_m ^ 2.0) / (k ^ 4.0))));
	elseif (t_m <= 3.3e+96)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(k / l_m)) / Float64(l_m / Float64(2.0 * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k / (Float64(cbrt(l_m) / t_m) ^ 3.0)) / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-88], N[(2.0 / N[(t$95$m / N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+96], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k / N[Power[N[(N[Power[l$95$m, 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.1999999999999995e-88

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 54.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      2. pow2 19.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* 22.8%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}} \]

   7. Simplified 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   8. Taylor expanded in k around inf 38.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. associate-*l/ 38.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   10. Simplified 38.5%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   11. Taylor expanded in k around 0 55.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. *-commutative 55.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

       2. associate-/l* 56.5%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   13. Simplified 56.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   if 6.1999999999999995e-88 < t < 3.29999999999999984e96

   1. Initial program 74.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 74.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 74.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 74.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 76.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 76.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 61.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 61.3%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 61.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 66.4%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 66.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 68.7%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{{t}^{3}}} \cdot \left(k \cdot 2\right)}{\ell}}} \]

       2. associate-/r/ 76.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}{\ell}} \]

   13. Applied egg-rr 76.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-/l* 78.6%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\ell} \cdot {t}^{3}}{\frac{\ell}{k \cdot 2}}}} \]

       2. *-commutative 78.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot 2}}} \]

   15. Simplified 78.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot 2}}}} \]

   if 3.29999999999999984e96 < t

   1. Initial program 54.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 52.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 52.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 63.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 63.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 63.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 63.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 63.5%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\left(\sqrt[3]{\frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       2. pow2 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{{\left(\sqrt[3]{\frac{\ell}{{t}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       3. cbrt-div 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       4. unpow3 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       5. add-cbrt-cube 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{\color{blue}{t}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       6. cbrt-div 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{t}^{3}}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       7. unpow3 63.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       8. add-cbrt-cube 71.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{t}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   13. Applied egg-rr 71.5%

       \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{t}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   14. Step-by-step derivation

       1. pow-plus 71.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\left(2 + 1\right)}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       2. metadata-eval 71.5%

          \[\leadsto \frac{2}{\frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{\color{blue}{3}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   15. Simplified 71.5%

       \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{2 \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\frac{k}{{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3}}}{\ell}}\\ \end{array} \]

Alternative 15: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{t_m}{\frac{{l_m}^{2}}{{k}^{4}}}}\\ \mathbf{elif}\;t_m \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{{t_m}^{3} \cdot \frac{k}{l_m}}{\frac{l_m}{2 \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{{\left(t_m \cdot \sqrt[3]{\frac{k}{l_m}}\right)}^{3}}{l_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.6e-89)
    (/ 2.0 (/ t_m (/ (pow l_m 2.0) (pow k 4.0))))
    (if (<= t_m 3.3e+96)
      (/ 2.0 (/ (* (pow t_m 3.0) (/ k l_m)) (/ l_m (* 2.0 k))))
      (/ 2.0 (* (* 2.0 k) (/ (pow (* t_m (cbrt (/ k l_m))) 3.0) l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 5.6e-89) {
		tmp = 2.0 / (t_m / (pow(l_m, 2.0) / pow(k, 4.0)));
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (pow((t_m * cbrt((k / l_m))), 3.0) / l_m));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 5.6e-89) {
		tmp = 2.0 / (t_m / (Math.pow(l_m, 2.0) / Math.pow(k, 4.0)));
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.pow((t_m * Math.cbrt((k / l_m))), 3.0) / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 5.6e-89)
		tmp = Float64(2.0 / Float64(t_m / Float64((l_m ^ 2.0) / (k ^ 4.0))));
	elseif (t_m <= 3.3e+96)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(k / l_m)) / Float64(l_m / Float64(2.0 * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64((Float64(t_m * cbrt(Float64(k / l_m))) ^ 3.0) / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.6e-89], N[(2.0 / N[(t$95$m / N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+96], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Power[N[(k / l$95$m), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.5999999999999998e-89

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 54.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      2. pow2 19.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* 22.8%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}} \]

   7. Simplified 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   8. Taylor expanded in k around inf 38.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. associate-*l/ 38.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   10. Simplified 38.5%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   11. Taylor expanded in k around 0 55.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. *-commutative 55.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

       2. associate-/l* 56.5%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   13. Simplified 56.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   if 5.5999999999999998e-89 < t < 3.29999999999999984e96

   1. Initial program 74.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 74.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 74.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 74.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 74.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 74.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 74.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 76.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 76.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 61.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 61.3%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 61.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 66.4%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 66.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 68.7%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{{t}^{3}}} \cdot \left(k \cdot 2\right)}{\ell}}} \]

       2. associate-/r/ 76.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}{\ell}} \]

   13. Applied egg-rr 76.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-/l* 78.6%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\ell} \cdot {t}^{3}}{\frac{\ell}{k \cdot 2}}}} \]

       2. *-commutative 78.6%

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot 2}}} \]

   15. Simplified 78.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot 2}}}} \]

   if 3.29999999999999984e96 < t

   1. Initial program 54.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 54.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 52.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 52.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 63.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 63.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 63.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 63.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 63.5%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}} \cdot \sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}}\right) \cdot \sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       2. pow3 63.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{\frac{k}{\frac{\ell}{{t}^{3}}}}\right)}^{3}}}{\ell} \cdot \left(k \cdot 2\right)} \]

       3. associate-/r/ 63.2%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\color{blue}{\frac{k}{\ell} \cdot {t}^{3}}}\right)}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

       4. cbrt-prod 63.2%

          \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{{t}^{3}}\right)}}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

       5. unpow3 63.2%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}\right)}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

       6. add-cbrt-cube 82.0%

          \[\leadsto \frac{2}{\frac{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \color{blue}{t}\right)}^{3}}{\ell} \cdot \left(k \cdot 2\right)} \]

   13. Applied egg-rr 82.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{\frac{k}{\ell}} \cdot t\right)}^{3}}}{\ell} \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{2 \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{{\left(t \cdot \sqrt[3]{\frac{k}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]

Alternative 16: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \frac{{l_m}^{2}}{t_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t_m}^{3} \cdot \frac{k}{l_m}}{\frac{l_m}{2 \cdot k}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-89)
    (* 2.0 (/ (pow l_m 2.0) (* t_m (pow k 4.0))))
    (/ 2.0 (/ (* (pow t_m 3.0) (/ k l_m)) (/ l_m (* 2.0 k)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 8e-89) {
		tmp = 2.0 * (pow(l_m, 2.0) / (t_m * pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d-89) then
        tmp = 2.0d0 * ((l_m ** 2.0d0) / (t_m * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / (((t_m ** 3.0d0) * (k / l_m)) / (l_m / (2.0d0 * k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 8e-89) {
		tmp = 2.0 * (Math.pow(l_m, 2.0) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 8e-89:
		tmp = 2.0 * (math.pow(l_m, 2.0) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 8e-89)
		tmp = Float64(2.0 * Float64((l_m ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(k / l_m)) / Float64(l_m / Float64(2.0 * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 8e-89)
		tmp = 2.0 * ((l_m ^ 2.0) / (t_m * (k ^ 4.0)));
	else
		tmp = 2.0 / (((t_m ^ 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-89], N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000031e-89

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 42.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Taylor expanded in k around inf 55.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   if 8.00000000000000031e-89 < t

   1. Initial program 64.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 64.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 64.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 64.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 68.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 68.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 68.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 63.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 63.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 68.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 68.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 68.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 68.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 70.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 70.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 62.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 62.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 62.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 65.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 65.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 66.1%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{{t}^{3}}} \cdot \left(k \cdot 2\right)}{\ell}}} \]

       2. associate-/r/ 69.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}{\ell}} \]

   13. Applied egg-rr 69.7%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-/l* 70.9%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\ell} \cdot {t}^{3}}{\frac{\ell}{k \cdot 2}}}} \]

       2. *-commutative 70.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot 2}}} \]

   15. Simplified 70.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot 2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{2 \cdot k}}}\\ \end{array} \]

Alternative 17: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.18 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{t_m}{\frac{{l_m}^{2}}{{k}^{4}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t_m}^{3} \cdot \frac{k}{l_m}}{\frac{l_m}{2 \cdot k}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.18e-88)
    (/ 2.0 (/ t_m (/ (pow l_m 2.0) (pow k 4.0))))
    (/ 2.0 (/ (* (pow t_m 3.0) (/ k l_m)) (/ l_m (* 2.0 k)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.18e-88) {
		tmp = 2.0 / (t_m / (pow(l_m, 2.0) / pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.18d-88) then
        tmp = 2.0d0 / (t_m / ((l_m ** 2.0d0) / (k ** 4.0d0)))
    else
        tmp = 2.0d0 / (((t_m ** 3.0d0) * (k / l_m)) / (l_m / (2.0d0 * k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.18e-88) {
		tmp = 2.0 / (t_m / (Math.pow(l_m, 2.0) / Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 1.18e-88:
		tmp = 2.0 / (t_m / (math.pow(l_m, 2.0) / math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.18e-88)
		tmp = Float64(2.0 / Float64(t_m / Float64((l_m ^ 2.0) / (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(k / l_m)) / Float64(l_m / Float64(2.0 * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.18e-88)
		tmp = 2.0 / (t_m / ((l_m ^ 2.0) / (k ^ 4.0)));
	else
		tmp = 2.0 / (((t_m ^ 3.0) * (k / l_m)) / (l_m / (2.0 * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.18e-88], N[(2.0 / N[(t$95$m / N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.18000000000000004e-88

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 48.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 54.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 40.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 54.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 54.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 19.9%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      2. pow2 19.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r* 22.8%

         \[\leadsto \frac{2}{{\left(\frac{\sqrt{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}} \]

   7. Simplified 22.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\sqrt{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   8. Taylor expanded in k around inf 38.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. associate-*l/ 38.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   10. Simplified 38.5%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

   11. Taylor expanded in k around 0 55.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. *-commutative 55.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

       2. associate-/l* 56.5%

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   13. Simplified 56.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   if 1.18000000000000004e-88 < t

   1. Initial program 64.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 64.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 64.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 64.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 68.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 68.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 68.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 63.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 63.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 68.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 68.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 68.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 68.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 70.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 70.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 62.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 62.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 62.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 63.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 65.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 65.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 66.1%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{{t}^{3}}} \cdot \left(k \cdot 2\right)}{\ell}}} \]

       2. associate-/r/ 69.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}{\ell}} \]

   13. Applied egg-rr 69.7%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}{\ell}}} \]
   14. Step-by-step derivation

       1. associate-/l* 70.9%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\ell} \cdot {t}^{3}}{\frac{\ell}{k \cdot 2}}}} \]

       2. *-commutative 70.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot 2}}} \]

   15. Simplified 70.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot 2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{2 \cdot k}}}\\ \end{array} \]

Alternative 18: 60.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\left(2 \cdot k\right) \cdot \frac{\frac{k}{l_m \cdot {t_m}^{-3}}}{l_m}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 k) (/ (/ k (* l_m (pow t_m -3.0))) l_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / ((2.0 * k) * ((k / (l_m * pow(t_m, -3.0))) / l_m)));
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * k) * ((k / (l_m * (t_m ** (-3.0d0)))) / l_m)))
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / ((2.0 * k) * ((k / (l_m * Math.pow(t_m, -3.0))) / l_m)));
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (2.0 / ((2.0 * k) * ((k / (l_m * math.pow(t_m, -3.0))) / l_m)))

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k / Float64(l_m * (t_m ^ -3.0))) / l_m))))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (2.0 / ((2.0 * k) * ((k / (l_m * (t_m ^ -3.0))) / l_m)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k / N[(l$95$m * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. *-commutative 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. *-commutative 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 58.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 58.3%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 58.3%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 47.4%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 47.4%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 58.3%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 58.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 58.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   12. +-commutative 58.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*l/ 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

5. Applied egg-rr 59.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

6. Taylor expanded in k around 0 55.7%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
7. Step-by-step derivation

   1. *-commutative 55.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

8. Simplified 55.7%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

9. Taylor expanded in k around 0 57.1%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
10. Step-by-step derivation

    1. associate-/l* 58.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

11. Simplified 58.7%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
12. Step-by-step derivation

    1. expm1-log1p-u 42.7%

       \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)}}}{\ell} \cdot \left(k \cdot 2\right)} \]

    2. expm1-udef 37.7%

       \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1}}}{\ell} \cdot \left(k \cdot 2\right)} \]

    3. div-inv 37.7%

       \[\leadsto \frac{2}{\frac{\frac{k}{e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1}}{\ell} \cdot \left(k \cdot 2\right)} \]

    4. pow-flip 37.7%

       \[\leadsto \frac{2}{\frac{\frac{k}{e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}}{\ell} \cdot \left(k \cdot 2\right)} \]

    5. metadata-eval 37.7%

       \[\leadsto \frac{2}{\frac{\frac{k}{e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1}}{\ell} \cdot \left(k \cdot 2\right)} \]

13. Applied egg-rr 37.7%

    \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1}}}{\ell} \cdot \left(k \cdot 2\right)} \]
14. Step-by-step derivation

    1. expm1-def 44.0%

       \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)}}}{\ell} \cdot \left(k \cdot 2\right)} \]

    2. expm1-log1p 60.0%

       \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\ell \cdot {t}^{-3}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

15. Simplified 60.0%

    \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\ell \cdot {t}^{-3}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

16. Final simplification 60.0%

    \[\leadsto \frac{2}{\left(2 \cdot k\right) \cdot \frac{\frac{k}{\ell \cdot {t}^{-3}}}{\ell}} \]

Alternative 19: 61.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{{t_m}^{3} \cdot \frac{k}{l_m}}{\frac{l_m}{2 \cdot k}}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (/ (* (pow t_m 3.0) (/ k l_m)) (/ l_m (* 2.0 k))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / ((pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k))));
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m ** 3.0d0) * (k / l_m)) / (l_m / (2.0d0 * k))))
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / ((Math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k))));
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (2.0 / ((math.pow(t_m, 3.0) * (k / l_m)) / (l_m / (2.0 * k))))

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(k / l_m)) / Float64(l_m / Float64(2.0 * k)))))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (2.0 / (((t_m ^ 3.0) * (k / l_m)) / (l_m / (2.0 * k))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. *-commutative 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. *-commutative 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 58.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 58.3%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 58.3%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 47.4%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 47.4%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 58.3%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 58.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 58.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   12. +-commutative 58.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*l/ 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

5. Applied egg-rr 59.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

6. Taylor expanded in k around 0 55.7%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]
7. Step-by-step derivation

   1. *-commutative 55.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

8. Simplified 55.7%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

9. Taylor expanded in k around 0 57.1%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot {t}^{3}}{\ell}}}{\ell} \cdot \left(k \cdot 2\right)} \]
10. Step-by-step derivation

    1. associate-/l* 58.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]

11. Simplified 58.7%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{t}^{3}}}}}{\ell} \cdot \left(k \cdot 2\right)} \]
12. Step-by-step derivation

    1. associate-*l/ 57.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{{t}^{3}}} \cdot \left(k \cdot 2\right)}{\ell}}} \]

    2. associate-/r/ 59.1%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}{\ell}} \]

13. Applied egg-rr 59.1%

    \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}{\ell}}} \]
14. Step-by-step derivation

    1. associate-/l* 60.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\ell} \cdot {t}^{3}}{\frac{\ell}{k \cdot 2}}}} \]

    2. *-commutative 60.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot 2}}} \]

15. Simplified 60.9%

    \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot 2}}}} \]

16. Final simplification 60.9%

    \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \frac{k}{\ell}}{\frac{\ell}{2 \cdot k}}} \]

Reproduce

herbie shell --seed 2023333 
(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))))