Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 91.0%

Time: 34.3s

Alternatives: 21

Speedup: 3.8×

• 27.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{l\_m}}\right)}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\left(\sqrt[3]{l\_m}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

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 4e-15)
    (/
     2.0
     (/
      (* (pow (/ k (sqrt l_m)) 2.0) (/ (* t_m (pow (sin k) 2.0)) (cos k)))
      l_m))
    (/
     2.0
     (pow
      (*
       (* (cbrt (sin k)) (* t_m (pow (cbrt l_m) -2.0)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      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 tmp;
	if (t_m <= 4e-15) {
		tmp = 2.0 / ((pow((k / sqrt(l_m)), 2.0) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l_m);
	} else {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m * pow(cbrt(l_m), -2.0))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 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 tmp;
	if (t_m <= 4e-15) {
		tmp = 2.0 / ((Math.pow((k / Math.sqrt(l_m)), 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l_m);
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m * Math.pow(Math.cbrt(l_m), -2.0))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 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)
	tmp = 0.0
	if (t_m <= 4e-15)
		tmp = Float64(2.0 / Float64(Float64((Float64(k / sqrt(l_m)) ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m));
	else
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * Float64(t_m * (cbrt(l_m) ^ -2.0))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 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_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-15], N[(2.0 / N[(N[(N[Power[N[(k / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000003e-15

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. sqr-neg 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-/r* 47.9%

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

      5. distribute-rgt-in 47.9%

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

      6. unpow2 47.9%

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

      7. times-frac 33.0%

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

      8. sqr-neg 33.0%

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

      9. times-frac 47.9%

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

      10. unpow2 47.9%

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

      11. distribute-rgt-in 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*l* 45.6%

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

      2. associate-*l/ 48.2%

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

      3. associate-*r* 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in t around 0 67.3%

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

      1. times-frac 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 32.1%

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

       2. sqrt-div 32.1%

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

       3. unpow2 32.1%

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

       4. sqrt-prod 15.4%

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

       5. add-sqr-sqrt 21.7%

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

       6. sqrt-div 21.7%

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

       7. unpow2 21.7%

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

       8. sqrt-prod 17.0%

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

       9. add-sqr-sqrt 33.7%

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

   11. Applied egg-rr 33.7%

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

       1. unpow2 33.7%

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

   13. Simplified 33.7%

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

   if 4.0000000000000003e-15 < t

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. sqr-neg 67.7%

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

      3. sqr-neg 67.7%

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

      4. associate-/r* 68.6%

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

      5. distribute-rgt-in 68.6%

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

      6. unpow2 68.6%

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

      7. times-frac 56.5%

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

      8. sqr-neg 56.5%

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

      9. times-frac 68.6%

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

      10. unpow2 68.6%

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

      11. distribute-rgt-in 68.6%

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

   3. Simplified 68.6%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. *-commutative 68.4%

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

      4. cbrt-prod 68.4%

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

      5. associate-/l/ 67.5%

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

      6. cbrt-div 67.5%

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

      7. rem-cbrt-cube 74.7%

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

      8. cbrt-unprod 93.6%

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

      9. pow2 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. add-cube-cbrt 93.2%

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

      2. pow3 93.3%

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

      3. div-inv 93.3%

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

      4. pow-flip 93.3%

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

      5. metadata-eval 93.3%

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

   8. Applied egg-rr 93.3%

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

      1. add-cube-cbrt 93.4%

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

      2. pow3 93.4%

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

      3. cbrt-prod 93.1%

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

      4. rem-cbrt-cube 96.9%

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

      5. unpow3 96.9%

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

      6. add-cube-cbrt 97.1%

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

   10. Applied egg-rr 97.1%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{\ell}}\right)}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{l\_m}}\right)}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left(t\_2 \cdot t\_2\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (* (pow t_m 0.75) (sqrt k)) -1.0)))
   (*
    t_s
    (if (<= t_m 2.8e-15)
      (/
       2.0
       (/
        (* (pow (/ k (sqrt l_m)) 2.0) (/ (* t_m (pow (sin k) 2.0)) (cos k)))
        l_m))
      (if (<= t_m 3.5e+101)
        (/
         2.0
         (*
          (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
          (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
        (pow (* l_m (* t_2 t_2)) 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((pow(t_m, 0.75) * sqrt(k)), -1.0);
	double tmp;
	if (t_m <= 2.8e-15) {
		tmp = 2.0 / ((pow((k / sqrt(l_m)), 2.0) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l_m);
	} else if (t_m <= 3.5e+101) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = pow((l_m * (t_2 * t_2)), 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 = ((t_m ** 0.75d0) * sqrt(k)) ** (-1.0d0)
    if (t_m <= 2.8d-15) then
        tmp = 2.0d0 / ((((k / sqrt(l_m)) ** 2.0d0) * ((t_m * (sin(k) ** 2.0d0)) / cos(k))) / l_m)
    else if (t_m <= 3.5d+101) then
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * ((sin(k) * ((t_m ** 3.0d0) / l_m)) / l_m))
    else
        tmp = (l_m * (t_2 * t_2)) ** 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 = Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -1.0);
	double tmp;
	if (t_m <= 2.8e-15) {
		tmp = 2.0 / ((Math.pow((k / Math.sqrt(l_m)), 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l_m);
	} else if (t_m <= 3.5e+101) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = Math.pow((l_m * (t_2 * t_2)), 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((math.pow(t_m, 0.75) * math.sqrt(k)), -1.0)
	tmp = 0
	if t_m <= 2.8e-15:
		tmp = 2.0 / ((math.pow((k / math.sqrt(l_m)), 2.0) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k))) / l_m)
	elif t_m <= 3.5e+101:
		tmp = 2.0 / ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * ((math.sin(k) * (math.pow(t_m, 3.0) / l_m)) / l_m))
	else:
		tmp = math.pow((l_m * (t_2 * t_2)), 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((t_m ^ 0.75) * sqrt(k)) ^ -1.0
	tmp = 0.0
	if (t_m <= 2.8e-15)
		tmp = Float64(2.0 / Float64(Float64((Float64(k / sqrt(l_m)) ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m));
	elseif (t_m <= 3.5e+101)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	else
		tmp = Float64(l_m * Float64(t_2 * t_2)) ^ 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 = ((t_m ^ 0.75) * sqrt(k)) ^ -1.0;
	tmp = 0.0;
	if (t_m <= 2.8e-15)
		tmp = 2.0 / ((((k / sqrt(l_m)) ^ 2.0) * ((t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m);
	elseif (t_m <= 3.5e+101)
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * ((sin(k) * ((t_m ^ 3.0) / l_m)) / l_m));
	else
		tmp = (l_m * (t_2 * t_2)) ^ 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[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.8e-15], N[(2.0 / N[(N[(N[Power[N[(k / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+101], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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], N[Power[N[(l$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.80000000000000014e-15

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. sqr-neg 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-/r* 47.9%

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

      5. distribute-rgt-in 47.9%

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

      6. unpow2 47.9%

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

      7. times-frac 33.0%

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

      8. sqr-neg 33.0%

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

      9. times-frac 47.9%

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

      10. unpow2 47.9%

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

      11. distribute-rgt-in 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*l* 45.6%

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

      2. associate-*l/ 48.2%

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

      3. associate-*r* 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in t around 0 67.3%

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

      1. times-frac 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 32.1%

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

       2. sqrt-div 32.1%

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

       3. unpow2 32.1%

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

       4. sqrt-prod 15.4%

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

       5. add-sqr-sqrt 21.7%

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

       6. sqrt-div 21.7%

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

       7. unpow2 21.7%

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

       8. sqrt-prod 17.0%

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

       9. add-sqr-sqrt 33.7%

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

   11. Applied egg-rr 33.7%

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

       1. unpow2 33.7%

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

   13. Simplified 33.7%

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

   if 2.80000000000000014e-15 < t < 3.50000000000000023e101

   1. Initial program 95.7%

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

      1. associate-*l* 95.7%

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

      2. sqr-neg 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-/r* 95.7%

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

      5. distribute-rgt-in 95.7%

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

      6. unpow2 95.7%

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

      7. times-frac 88.6%

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

      8. sqr-neg 88.6%

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

      9. times-frac 95.7%

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

      10. unpow2 95.7%

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

      11. distribute-rgt-in 95.7%

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

   3. Simplified 95.7%

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

      1. associate-*l/ 95.7%

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

   6. Applied egg-rr 95.7%

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

   if 3.50000000000000023e101 < t

   1. Initial program 52.6%

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

      1. associate-/r* 52.6%

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

      2. sqr-neg 52.6%

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

      3. associate-*l* 46.0%

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

      4. sqr-neg 46.0%

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

      5. associate-/r* 47.2%

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

      6. associate-+l+ 47.2%

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

      7. unpow2 47.2%

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

      8. times-frac 32.4%

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

      9. sqr-neg 32.4%

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

      10. times-frac 47.2%

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

      11. unpow2 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in k around 0 46.0%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-udef 46.0%

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

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 75.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 76.0%

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

   9. Simplified 76.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 76.1%

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

   11. Applied egg-rr 76.1%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 76.1%

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

       2. add-sqr-sqrt 31.5%

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

       3. unpow-prod-down 31.5%

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

       4. *-commutative 31.5%

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

       5. sqrt-prod 31.5%

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

       6. sqrt-pow1 31.5%

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

       7. metadata-eval 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 34.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 34.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 34.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{\ell}}\right)}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{l\_m}}\right)}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

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 4.8e-15)
    (/
     2.0
     (/
      (* (pow (/ k (sqrt l_m)) 2.0) (/ (* t_m (pow (sin k) 2.0)) (cos k)))
      l_m))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l_m) 2.0))) 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 tmp;
	if (t_m <= 4.8e-15) {
		tmp = 2.0 / ((pow((k / sqrt(l_m)), 2.0) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l_m);
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l_m), 2.0))), 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 tmp;
	if (t_m <= 4.8e-15) {
		tmp = 2.0 / ((Math.pow((k / Math.sqrt(l_m)), 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l_m);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l_m), 2.0))), 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)
	tmp = 0.0
	if (t_m <= 4.8e-15)
		tmp = Float64(2.0 / Float64(Float64((Float64(k / sqrt(l_m)) ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l_m) ^ 2.0))) ^ 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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-15], N[(2.0 / N[(N[(N[Power[N[(k / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.7999999999999999e-15

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. sqr-neg 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-/r* 47.9%

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

      5. distribute-rgt-in 47.9%

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

      6. unpow2 47.9%

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

      7. times-frac 33.0%

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

      8. sqr-neg 33.0%

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

      9. times-frac 47.9%

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

      10. unpow2 47.9%

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

      11. distribute-rgt-in 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*l* 45.6%

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

      2. associate-*l/ 48.2%

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

      3. associate-*r* 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in t around 0 67.3%

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

      1. times-frac 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 32.1%

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

       2. sqrt-div 32.1%

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

       3. unpow2 32.1%

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

       4. sqrt-prod 15.4%

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

       5. add-sqr-sqrt 21.7%

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

       6. sqrt-div 21.7%

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

       7. unpow2 21.7%

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

       8. sqrt-prod 17.0%

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

       9. add-sqr-sqrt 33.7%

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

   11. Applied egg-rr 33.7%

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

       1. unpow2 33.7%

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

   13. Simplified 33.7%

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

   if 4.7999999999999999e-15 < t

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. sqr-neg 67.7%

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

      3. sqr-neg 67.7%

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

      4. associate-/r* 68.6%

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

      5. distribute-rgt-in 68.6%

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

      6. unpow2 68.6%

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

      7. times-frac 56.5%

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

      8. sqr-neg 56.5%

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

      9. times-frac 68.6%

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

      10. unpow2 68.6%

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

      11. distribute-rgt-in 68.6%

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

   3. Simplified 68.6%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. *-commutative 68.4%

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

      4. cbrt-prod 68.4%

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

      5. associate-/l/ 67.5%

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

      6. cbrt-div 67.5%

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

      7. rem-cbrt-cube 74.7%

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

      8. cbrt-unprod 93.6%

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

      9. pow2 93.6%

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

   6. Applied egg-rr 93.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{\ell}}\right)}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{l\_m}}\right)}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\frac{\sqrt[3]{\sin k}}{\frac{{\left(\sqrt[3]{l\_m}\right)}^{2}}{t\_m}}\right)}^{3}}\\ \end{array} \end{array} \]

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 1.55e-15)
    (/
     2.0
     (/
      (* (pow (/ k (sqrt l_m)) 2.0) (/ (* t_m (pow (sin k) 2.0)) (cos k)))
      l_m))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (/ (cbrt (sin k)) (/ (pow (cbrt l_m) 2.0) t_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 tmp;
	if (t_m <= 1.55e-15) {
		tmp = 2.0 / ((pow((k / sqrt(l_m)), 2.0) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l_m);
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((cbrt(sin(k)) / (pow(cbrt(l_m), 2.0) / t_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 tmp;
	if (t_m <= 1.55e-15) {
		tmp = 2.0 / ((Math.pow((k / Math.sqrt(l_m)), 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l_m);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((Math.cbrt(Math.sin(k)) / (Math.pow(Math.cbrt(l_m), 2.0) / t_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)
	tmp = 0.0
	if (t_m <= 1.55e-15)
		tmp = Float64(2.0 / Float64(Float64((Float64(k / sqrt(l_m)) ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(cbrt(sin(k)) / Float64((cbrt(l_m) ^ 2.0) / t_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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.55e-15], N[(2.0 / N[(N[(N[Power[N[(k / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5499999999999999e-15

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. sqr-neg 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-/r* 47.9%

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

      5. distribute-rgt-in 47.9%

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

      6. unpow2 47.9%

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

      7. times-frac 33.0%

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

      8. sqr-neg 33.0%

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

      9. times-frac 47.9%

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

      10. unpow2 47.9%

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

      11. distribute-rgt-in 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*l* 45.6%

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

      2. associate-*l/ 48.2%

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

      3. associate-*r* 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in t around 0 67.3%

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

      1. times-frac 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 32.1%

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

       2. sqrt-div 32.1%

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

       3. unpow2 32.1%

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

       4. sqrt-prod 15.4%

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

       5. add-sqr-sqrt 21.7%

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

       6. sqrt-div 21.7%

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

       7. unpow2 21.7%

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

       8. sqrt-prod 17.0%

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

       9. add-sqr-sqrt 33.7%

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

   11. Applied egg-rr 33.7%

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

       1. unpow2 33.7%

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

   13. Simplified 33.7%

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

   if 1.5499999999999999e-15 < t

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. sqr-neg 67.7%

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

      3. sqr-neg 67.7%

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

      4. associate-/r* 68.6%

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

      5. distribute-rgt-in 68.6%

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

      6. unpow2 68.6%

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

      7. times-frac 56.5%

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

      8. sqr-neg 56.5%

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

      9. times-frac 68.6%

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

      10. unpow2 68.6%

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

      11. distribute-rgt-in 68.6%

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

   3. Simplified 68.6%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. *-commutative 68.4%

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

      4. cbrt-prod 68.4%

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

      5. associate-/l/ 67.5%

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

      6. cbrt-div 67.5%

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

      7. rem-cbrt-cube 74.7%

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

      8. cbrt-unprod 93.6%

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

      9. pow2 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. associate-*r/ 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. associate-/l* 93.7%

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

   10. Simplified 93.7%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{\ell}}\right)}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{\sqrt[3]{\sin k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{l\_m}}\right)}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \]

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 6.8e-16)
    (/
     2.0
     (/
      (* (pow (/ k (sqrt l_m)) 2.0) (/ (* t_m (pow (sin k) 2.0)) (cos k)))
      l_m))
    (if (<= t_m 5.5e+102)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
      (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 <= 6.8e-16) {
		tmp = 2.0 / ((pow((k / sqrt(l_m)), 2.0) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l_m);
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 <= 6.8d-16) then
        tmp = 2.0d0 / ((((k / sqrt(l_m)) ** 2.0d0) * ((t_m * (sin(k) ** 2.0d0)) / cos(k))) / l_m)
    else if (t_m <= 5.5d+102) then
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * ((sin(k) * ((t_m ** 3.0d0) / l_m)) / l_m))
    else
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-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 <= 6.8e-16) {
		tmp = 2.0 / ((Math.pow((k / Math.sqrt(l_m)), 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l_m);
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -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 <= 6.8e-16:
		tmp = 2.0 / ((math.pow((k / math.sqrt(l_m)), 2.0) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k))) / l_m)
	elif t_m <= 5.5e+102:
		tmp = 2.0 / ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * ((math.sin(k) * (math.pow(t_m, 3.0) / l_m)) / l_m))
	else:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -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 <= 6.8e-16)
		tmp = Float64(2.0 / Float64(Float64((Float64(k / sqrt(l_m)) ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m));
	elseif (t_m <= 5.5e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -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 <= 6.8e-16)
		tmp = 2.0 / ((((k / sqrt(l_m)) ^ 2.0) * ((t_m * (sin(k) ^ 2.0)) / cos(k))) / l_m);
	elseif (t_m <= 5.5e+102)
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * ((sin(k) * ((t_m ^ 3.0) / l_m)) / l_m));
	else
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -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, 6.8e-16], N[(2.0 / N[(N[(N[Power[N[(k / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+102], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.8e-16

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. sqr-neg 45.4%

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

      3. sqr-neg 45.4%

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

      4. associate-/r* 47.9%

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

      5. distribute-rgt-in 47.9%

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

      6. unpow2 47.9%

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

      7. times-frac 33.0%

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

      8. sqr-neg 33.0%

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

      9. times-frac 47.9%

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

      10. unpow2 47.9%

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

      11. distribute-rgt-in 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*l* 45.6%

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

      2. associate-*l/ 48.2%

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

      3. associate-*r* 48.2%

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

   6. Applied egg-rr 48.2%

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

   7. Taylor expanded in t around 0 67.3%

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

      1. times-frac 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 32.1%

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

       2. sqrt-div 32.1%

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

       3. unpow2 32.1%

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

       4. sqrt-prod 15.4%

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

       5. add-sqr-sqrt 21.7%

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

       6. sqrt-div 21.7%

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

       7. unpow2 21.7%

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

       8. sqrt-prod 17.0%

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

       9. add-sqr-sqrt 33.7%

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

   11. Applied egg-rr 33.7%

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

       1. unpow2 33.7%

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

   13. Simplified 33.7%

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

   if 6.8e-16 < t < 5.49999999999999981e102

   1. Initial program 95.7%

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

      1. associate-*l* 95.7%

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

      2. sqr-neg 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-/r* 95.7%

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

      5. distribute-rgt-in 95.7%

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

      6. unpow2 95.7%

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

      7. times-frac 88.6%

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

      8. sqr-neg 88.6%

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

      9. times-frac 95.7%

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

      10. unpow2 95.7%

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

      11. distribute-rgt-in 95.7%

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

   3. Simplified 95.7%

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

      1. associate-*l/ 95.7%

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

   6. Applied egg-rr 95.7%

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

   if 5.49999999999999981e102 < t

   1. Initial program 52.6%

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

      1. associate-/r* 52.6%

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

      2. sqr-neg 52.6%

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

      3. associate-*l* 46.0%

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

      4. sqr-neg 46.0%

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

      5. associate-/r* 47.2%

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

      6. associate-+l+ 47.2%

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

      7. unpow2 47.2%

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

      8. times-frac 32.4%

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

      9. sqr-neg 32.4%

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

      10. times-frac 47.2%

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

      11. unpow2 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in k around 0 46.0%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-udef 46.0%

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

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 75.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 76.0%

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

   9. Simplified 76.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 76.1%

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

   11. Applied egg-rr 76.1%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 76.1%

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

       2. add-sqr-sqrt 31.5%

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

       3. unpow-prod-down 31.5%

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

       4. *-commutative 31.5%

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

       5. sqrt-prod 31.5%

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

       6. sqrt-pow1 31.5%

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

       7. metadata-eval 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 34.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 34.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 34.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 34.5%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 34.5%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 34.5%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\sqrt{\ell}}\right)}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\cos k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-201}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{l\_m}{\sin k}} \cdot \sqrt{t\_2}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{l\_m}{\frac{{k}^{2}}{l\_m} \cdot \left({\sin k}^{2} \cdot t\_2\right)}}}\\ \mathbf{elif}\;t\_m \leq 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (/ t_m (cos k))))
   (*
    t_s
    (if (<= t_m 1.6e-201)
      (/ 2.0 (pow (* (/ k (/ l_m (sin k))) (sqrt t_2)) 2.0))
      (if (<= t_m 4.6e-15)
        (/
         2.0
         (/ 1.0 (/ l_m (* (/ (pow k 2.0) l_m) (* (pow (sin k) 2.0) t_2)))))
        (if (<= t_m 1e+102)
          (/
           2.0
           (*
            (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
            (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
          (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 = t_m / cos(k);
	double tmp;
	if (t_m <= 1.6e-201) {
		tmp = 2.0 / pow(((k / (l_m / sin(k))) * sqrt(t_2)), 2.0);
	} else if (t_m <= 4.6e-15) {
		tmp = 2.0 / (1.0 / (l_m / ((pow(k, 2.0) / l_m) * (pow(sin(k), 2.0) * t_2))));
	} else if (t_m <= 1e+102) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 = t_m / cos(k)
    if (t_m <= 1.6d-201) then
        tmp = 2.0d0 / (((k / (l_m / sin(k))) * sqrt(t_2)) ** 2.0d0)
    else if (t_m <= 4.6d-15) then
        tmp = 2.0d0 / (1.0d0 / (l_m / (((k ** 2.0d0) / l_m) * ((sin(k) ** 2.0d0) * t_2))))
    else if (t_m <= 1d+102) then
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * ((sin(k) * ((t_m ** 3.0d0) / l_m)) / l_m))
    else
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-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 = t_m / Math.cos(k);
	double tmp;
	if (t_m <= 1.6e-201) {
		tmp = 2.0 / Math.pow(((k / (l_m / Math.sin(k))) * Math.sqrt(t_2)), 2.0);
	} else if (t_m <= 4.6e-15) {
		tmp = 2.0 / (1.0 / (l_m / ((Math.pow(k, 2.0) / l_m) * (Math.pow(Math.sin(k), 2.0) * t_2))));
	} else if (t_m <= 1e+102) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -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 = t_m / math.cos(k)
	tmp = 0
	if t_m <= 1.6e-201:
		tmp = 2.0 / math.pow(((k / (l_m / math.sin(k))) * math.sqrt(t_2)), 2.0)
	elif t_m <= 4.6e-15:
		tmp = 2.0 / (1.0 / (l_m / ((math.pow(k, 2.0) / l_m) * (math.pow(math.sin(k), 2.0) * t_2))))
	elif t_m <= 1e+102:
		tmp = 2.0 / ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * ((math.sin(k) * (math.pow(t_m, 3.0) / l_m)) / l_m))
	else:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -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(t_m / cos(k))
	tmp = 0.0
	if (t_m <= 1.6e-201)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l_m / sin(k))) * sqrt(t_2)) ^ 2.0));
	elseif (t_m <= 4.6e-15)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l_m / Float64(Float64((k ^ 2.0) / l_m) * Float64((sin(k) ^ 2.0) * t_2)))));
	elseif (t_m <= 1e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -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 = t_m / cos(k);
	tmp = 0.0;
	if (t_m <= 1.6e-201)
		tmp = 2.0 / (((k / (l_m / sin(k))) * sqrt(t_2)) ^ 2.0);
	elseif (t_m <= 4.6e-15)
		tmp = 2.0 / (1.0 / (l_m / (((k ^ 2.0) / l_m) * ((sin(k) ^ 2.0) * t_2))));
	elseif (t_m <= 1e+102)
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * ((sin(k) * ((t_m ^ 3.0) / l_m)) / l_m));
	else
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -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[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-201], N[(2.0 / N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e-15], N[(2.0 / N[(1.0 / N[(l$95$m / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+102], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.6000000000000001e-201

   1. Initial program 47.1%

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

      1. associate-*l* 47.1%

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

      2. sqr-neg 47.1%

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

      3. sqr-neg 47.1%

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

      4. associate-/r* 48.9%

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

      5. distribute-rgt-in 48.9%

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

      6. unpow2 48.9%

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

      7. times-frac 31.8%

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

      8. sqr-neg 31.8%

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

      9. times-frac 48.9%

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

      10. unpow2 48.9%

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

      11. distribute-rgt-in 48.9%

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

   3. Simplified 48.9%

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

      1. add-sqr-sqrt 16.0%

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

      2. pow2 16.0%

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

      3. associate-*l* 16.0%

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

      4. sqrt-prod 6.4%

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

      5. associate-/l/ 6.4%

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

      6. sqrt-div 6.4%

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

      7. sqrt-pow1 7.8%

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

      8. metadata-eval 7.8%

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

      9. sqrt-unprod 5.7%

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

      10. add-sqr-sqrt 7.8%

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

      11. associate-*r* 7.8%

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

   6. Applied egg-rr 7.8%

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

   7. Taylor expanded in k around inf 36.2%

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

      1. associate-/l* 36.2%

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

   9. Simplified 36.2%

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

   if 1.6000000000000001e-201 < t < 4.59999999999999981e-15

   1. Initial program 39.5%

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

      1. associate-*l* 39.5%

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

      2. sqr-neg 39.5%

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

      3. sqr-neg 39.5%

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

      4. associate-/r* 44.7%

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

      5. distribute-rgt-in 44.7%

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

      6. unpow2 44.7%

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

      7. times-frac 37.2%

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

      8. sqr-neg 37.2%

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

      9. times-frac 44.7%

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

      10. unpow2 44.7%

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

      11. distribute-rgt-in 44.7%

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

   3. Simplified 44.7%

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

      1. associate-*l* 42.4%

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

      2. associate-*l/ 47.5%

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

      3. associate-*r* 47.5%

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

   6. Applied egg-rr 47.5%

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

   7. Taylor expanded in t around 0 77.0%

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

      1. times-frac 77.9%

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

   9. Simplified 77.9%

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

       1. clear-num 78.0%

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

       2. inv-pow 78.0%

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

       3. associate-/l* 77.9%

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

   11. Applied egg-rr 77.9%

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

       1. unpow-1 77.9%

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

       2. associate-/r/ 78.0%

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

   13. Simplified 78.0%

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

   if 4.59999999999999981e-15 < t < 9.99999999999999977e101

   1. Initial program 95.7%

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

      1. associate-*l* 95.7%

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

      2. sqr-neg 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-/r* 95.7%

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

      5. distribute-rgt-in 95.7%

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

      6. unpow2 95.7%

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

      7. times-frac 88.6%

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

      8. sqr-neg 88.6%

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

      9. times-frac 95.7%

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

      10. unpow2 95.7%

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

      11. distribute-rgt-in 95.7%

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

   3. Simplified 95.7%

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

      1. associate-*l/ 95.7%

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

   6. Applied egg-rr 95.7%

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

   if 9.99999999999999977e101 < t

   1. Initial program 52.6%

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

      1. associate-/r* 52.6%

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

      2. sqr-neg 52.6%

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

      3. associate-*l* 46.0%

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

      4. sqr-neg 46.0%

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

      5. associate-/r* 47.2%

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

      6. associate-+l+ 47.2%

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

      7. unpow2 47.2%

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

      8. times-frac 32.4%

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

      9. sqr-neg 32.4%

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

      10. times-frac 47.2%

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

      11. unpow2 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in k around 0 46.0%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-udef 46.0%

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

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 75.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 76.0%

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

   9. Simplified 76.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 76.1%

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

   11. Applied egg-rr 76.1%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 76.1%

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

       2. add-sqr-sqrt 31.5%

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

       3. unpow-prod-down 31.5%

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

       4. *-commutative 31.5%

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

       5. sqrt-prod 31.5%

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

       6. sqrt-pow1 31.5%

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

       7. metadata-eval 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 34.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 34.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 34.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 34.5%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 34.5%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 34.5%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-201}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}}}\\ \mathbf{elif}\;t \leq 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{{\left(\frac{k}{\frac{l\_m}{\sin k}} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-163}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{l\_m} \cdot \frac{t\_m \cdot {k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (/ l_m (sin k))) (sqrt (/ t_m (cos k)))) 2.0))))
   (*
    t_s
    (if (<= t_m 2.6e-163)
      t_2
      (if (<= t_m 3.4e-101)
        (/ 2.0 (/ (* (/ (pow k 2.0) l_m) (/ (* t_m (pow k 2.0)) (cos k))) l_m))
        (if (<= t_m 8.5e-61)
          t_2
          (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 / (l_m / sin(k))) * sqrt((t_m / cos(k)))), 2.0);
	double tmp;
	if (t_m <= 2.6e-163) {
		tmp = t_2;
	} else if (t_m <= 3.4e-101) {
		tmp = 2.0 / (((pow(k, 2.0) / l_m) * ((t_m * pow(k, 2.0)) / cos(k))) / l_m);
	} else if (t_m <= 8.5e-61) {
		tmp = t_2;
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ** 2.0d0)
    if (t_m <= 2.6d-163) then
        tmp = t_2
    else if (t_m <= 3.4d-101) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l_m) * ((t_m * (k ** 2.0d0)) / cos(k))) / l_m)
    else if (t_m <= 8.5d-61) then
        tmp = t_2
    else
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-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 / (l_m / Math.sin(k))) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	double tmp;
	if (t_m <= 2.6e-163) {
		tmp = t_2;
	} else if (t_m <= 3.4e-101) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l_m) * ((t_m * Math.pow(k, 2.0)) / Math.cos(k))) / l_m);
	} else if (t_m <= 8.5e-61) {
		tmp = t_2;
	} else {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -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 / (l_m / math.sin(k))) * math.sqrt((t_m / math.cos(k)))), 2.0)
	tmp = 0
	if t_m <= 2.6e-163:
		tmp = t_2
	elif t_m <= 3.4e-101:
		tmp = 2.0 / (((math.pow(k, 2.0) / l_m) * ((t_m * math.pow(k, 2.0)) / math.cos(k))) / l_m)
	elif t_m <= 8.5e-61:
		tmp = t_2
	else:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -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(Float64(k / Float64(l_m / sin(k))) * sqrt(Float64(t_m / cos(k)))) ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.6e-163)
		tmp = t_2;
	elseif (t_m <= 3.4e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l_m) * Float64(Float64(t_m * (k ^ 2.0)) / cos(k))) / l_m));
	elseif (t_m <= 8.5e-61)
		tmp = t_2;
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -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 / (l_m / sin(k))) * sqrt((t_m / cos(k)))) ^ 2.0);
	tmp = 0.0;
	if (t_m <= 2.6e-163)
		tmp = t_2;
	elseif (t_m <= 3.4e-101)
		tmp = 2.0 / ((((k ^ 2.0) / l_m) * ((t_m * (k ^ 2.0)) / cos(k))) / l_m);
	elseif (t_m <= 8.5e-61)
		tmp = t_2;
	else
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -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[(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]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-163], t$95$2, If[LessEqual[t$95$m, 3.4e-101], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e-61], t$95$2, N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.60000000000000002e-163 or 3.39999999999999989e-101 < t < 8.50000000000000016e-61

   1. Initial program 45.9%

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

      1. associate-*l* 45.9%

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

      2. sqr-neg 45.9%

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

      3. sqr-neg 45.9%

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

      4. associate-/r* 48.1%

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

      5. distribute-rgt-in 48.1%

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

      6. unpow2 48.1%

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

      7. times-frac 30.9%

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

      8. sqr-neg 30.9%

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

      9. times-frac 48.1%

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

      10. unpow2 48.1%

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

      11. distribute-rgt-in 48.1%

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

   3. Simplified 48.1%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

      3. associate-*l* 18.1%

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

      4. sqrt-prod 9.6%

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

      5. associate-/l/ 9.5%

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

      6. sqrt-div 9.6%

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

      7. sqrt-pow1 11.0%

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

      8. metadata-eval 11.0%

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

      9. sqrt-unprod 8.3%

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

      10. add-sqr-sqrt 12.2%

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

      11. associate-*r* 12.2%

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

   6. Applied egg-rr 12.2%

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

   7. Taylor expanded in k around inf 40.5%

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

      1. associate-/l* 40.5%

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

   9. Simplified 40.5%

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

   if 2.60000000000000002e-163 < t < 3.39999999999999989e-101

   1. Initial program 27.3%

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

      1. associate-*l* 27.3%

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

      2. sqr-neg 27.3%

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

      3. sqr-neg 27.3%

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

      4. associate-/r* 37.3%

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

      5. distribute-rgt-in 37.3%

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

      6. unpow2 37.3%

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

      7. times-frac 37.3%

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

      8. sqr-neg 37.3%

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

      9. times-frac 37.3%

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

      10. unpow2 37.3%

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

      11. distribute-rgt-in 37.3%

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

   3. Simplified 37.3%

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

      1. associate-*l* 37.3%

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

      2. associate-*l/ 37.3%

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

      3. associate-*r* 37.3%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 83.0%

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

      1. times-frac 83.0%

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

   9. Simplified 83.0%

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

   10. Taylor expanded in k around 0 83.0%

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

   if 8.50000000000000016e-61 < t

   1. Initial program 65.9%

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

      1. associate-/r* 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 62.3%

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

      4. sqr-neg 62.3%

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

      5. associate-/r* 62.9%

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

      6. associate-+l+ 62.9%

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

      7. unpow2 62.9%

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

      8. times-frac 52.6%

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

      9. sqr-neg 52.6%

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

      10. times-frac 62.9%

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

      11. unpow2 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. expm1-log1p-u 53.4%

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

      2. expm1-udef 53.3%

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

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

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

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

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

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 71.6%

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

       2. add-sqr-sqrt 32.9%

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

       3. unpow-prod-down 32.9%

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

       4. *-commutative 32.9%

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

       5. sqrt-prod 31.7%

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

       6. sqrt-pow1 31.7%

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

       7. metadata-eval 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 33.4%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 33.4%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{t\_m}{\cos k}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{l\_m}{\sin k}} \cdot t\_2\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 9.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{l\_m} \cdot \frac{t\_m \cdot {k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 2.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \frac{k \cdot \sin k}{l\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (/ t_m (cos k)))))
   (*
    t_s
    (if (<= t_m 2.75e-163)
      (/ 2.0 (pow (* (/ k (/ l_m (sin k))) t_2) 2.0))
      (if (<= t_m 9.6e-96)
        (/ 2.0 (/ (* (/ (pow k 2.0) l_m) (/ (* t_m (pow k 2.0)) (cos k))) l_m))
        (if (<= t_m 2.6e-62)
          (/ 2.0 (pow (* t_2 (/ (* k (sin k)) l_m)) 2.0))
          (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 = sqrt((t_m / cos(k)));
	double tmp;
	if (t_m <= 2.75e-163) {
		tmp = 2.0 / pow(((k / (l_m / sin(k))) * t_2), 2.0);
	} else if (t_m <= 9.6e-96) {
		tmp = 2.0 / (((pow(k, 2.0) / l_m) * ((t_m * pow(k, 2.0)) / cos(k))) / l_m);
	} else if (t_m <= 2.6e-62) {
		tmp = 2.0 / pow((t_2 * ((k * sin(k)) / l_m)), 2.0);
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 = sqrt((t_m / cos(k)))
    if (t_m <= 2.75d-163) then
        tmp = 2.0d0 / (((k / (l_m / sin(k))) * t_2) ** 2.0d0)
    else if (t_m <= 9.6d-96) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l_m) * ((t_m * (k ** 2.0d0)) / cos(k))) / l_m)
    else if (t_m <= 2.6d-62) then
        tmp = 2.0d0 / ((t_2 * ((k * sin(k)) / l_m)) ** 2.0d0)
    else
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-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 = Math.sqrt((t_m / Math.cos(k)));
	double tmp;
	if (t_m <= 2.75e-163) {
		tmp = 2.0 / Math.pow(((k / (l_m / Math.sin(k))) * t_2), 2.0);
	} else if (t_m <= 9.6e-96) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l_m) * ((t_m * Math.pow(k, 2.0)) / Math.cos(k))) / l_m);
	} else if (t_m <= 2.6e-62) {
		tmp = 2.0 / Math.pow((t_2 * ((k * Math.sin(k)) / l_m)), 2.0);
	} else {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -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 = math.sqrt((t_m / math.cos(k)))
	tmp = 0
	if t_m <= 2.75e-163:
		tmp = 2.0 / math.pow(((k / (l_m / math.sin(k))) * t_2), 2.0)
	elif t_m <= 9.6e-96:
		tmp = 2.0 / (((math.pow(k, 2.0) / l_m) * ((t_m * math.pow(k, 2.0)) / math.cos(k))) / l_m)
	elif t_m <= 2.6e-62:
		tmp = 2.0 / math.pow((t_2 * ((k * math.sin(k)) / l_m)), 2.0)
	else:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -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 = sqrt(Float64(t_m / cos(k)))
	tmp = 0.0
	if (t_m <= 2.75e-163)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l_m / sin(k))) * t_2) ^ 2.0));
	elseif (t_m <= 9.6e-96)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l_m) * Float64(Float64(t_m * (k ^ 2.0)) / cos(k))) / l_m));
	elseif (t_m <= 2.6e-62)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(Float64(k * sin(k)) / l_m)) ^ 2.0));
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -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 = sqrt((t_m / cos(k)));
	tmp = 0.0;
	if (t_m <= 2.75e-163)
		tmp = 2.0 / (((k / (l_m / sin(k))) * t_2) ^ 2.0);
	elseif (t_m <= 9.6e-96)
		tmp = 2.0 / ((((k ^ 2.0) / l_m) * ((t_m * (k ^ 2.0)) / cos(k))) / l_m);
	elseif (t_m <= 2.6e-62)
		tmp = 2.0 / ((t_2 * ((k * sin(k)) / l_m)) ^ 2.0);
	else
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -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[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.75e-163], N[(2.0 / N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.6e-96], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.6e-62], N[(2.0 / N[Power[N[(t$95$2 * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.7499999999999999e-163

   1. Initial program 46.0%

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

      1. associate-*l* 46.0%

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

      2. sqr-neg 46.0%

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

      3. sqr-neg 46.0%

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

      4. associate-/r* 47.7%

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

      5. distribute-rgt-in 47.7%

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

      6. unpow2 47.7%

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

      7. times-frac 29.7%

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

      8. sqr-neg 29.7%

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

      9. times-frac 47.7%

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

      10. unpow2 47.7%

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

      11. distribute-rgt-in 47.7%

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

   3. Simplified 47.7%

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

      1. add-sqr-sqrt 17.0%

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

      2. pow2 17.0%

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

      3. associate-*l* 16.9%

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

      4. sqrt-prod 8.0%

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

      5. associate-/l/ 7.9%

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

      6. sqrt-div 7.9%

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

      7. sqrt-pow1 9.3%

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

      8. metadata-eval 9.3%

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

      9. sqrt-unprod 7.3%

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

      10. add-sqr-sqrt 10.6%

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

      11. associate-*r* 10.6%

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

   6. Applied egg-rr 10.6%

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

   7. Taylor expanded in k around inf 38.4%

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

      1. associate-/l* 38.4%

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

   9. Simplified 38.4%

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

   if 2.7499999999999999e-163 < t < 9.60000000000000076e-96

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. sqr-neg 33.2%

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

      3. sqr-neg 33.2%

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

      4. associate-/r* 42.4%

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

      5. distribute-rgt-in 42.4%

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

      6. unpow2 42.4%

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

      7. times-frac 42.3%

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

      8. sqr-neg 42.3%

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

      9. times-frac 42.4%

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

      10. unpow2 42.4%

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

      11. distribute-rgt-in 42.4%

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

   3. Simplified 42.4%

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

      1. associate-*l* 42.4%

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

      2. associate-*l/ 42.4%

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

      3. associate-*r* 42.4%

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

   6. Applied egg-rr 42.4%

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

   7. Taylor expanded in t around 0 84.3%

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

      1. times-frac 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in k around 0 84.3%

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

   if 9.60000000000000076e-96 < t < 2.5999999999999999e-62

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. sqr-neg 33.7%

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

      3. sqr-neg 33.7%

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

      4. associate-/r* 49.7%

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

      5. distribute-rgt-in 49.7%

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

      6. unpow2 49.7%

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

      7. times-frac 49.7%

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

      8. sqr-neg 49.7%

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

      9. times-frac 49.7%

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

      10. unpow2 49.7%

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

      11. distribute-rgt-in 49.7%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

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

      3. associate-*l* 34.2%

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

      4. sqrt-prod 34.2%

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

      5. associate-/l/ 33.3%

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

      6. sqrt-div 37.1%

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

      7. sqrt-pow1 37.1%

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

      8. metadata-eval 37.1%

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

      9. sqrt-unprod 34.5%

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

      10. add-sqr-sqrt 38.0%

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

      11. associate-*r* 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in k around inf 82.8%

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

   if 2.5999999999999999e-62 < t

   1. Initial program 65.9%

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

      1. associate-/r* 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 62.3%

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

      4. sqr-neg 62.3%

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

      5. associate-/r* 62.9%

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

      6. associate-+l+ 62.9%

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

      7. unpow2 62.9%

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

      8. times-frac 52.6%

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

      9. sqr-neg 52.6%

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

      10. times-frac 62.9%

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

      11. unpow2 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. expm1-log1p-u 53.4%

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

      2. expm1-udef 53.3%

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

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

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

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

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

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 71.6%

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

       2. add-sqr-sqrt 32.9%

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

       3. unpow-prod-down 32.9%

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

       4. *-commutative 32.9%

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

       5. sqrt-prod 31.7%

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

       6. sqrt-pow1 31.7%

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

       7. metadata-eval 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 33.4%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 33.4%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\frac{t\_m}{\cos k}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{l\_m}{\sin k}} \cdot t\_2\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{l\_m} \cdot \frac{t\_m \cdot {k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 3.25 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_2 \cdot \left(k \cdot \sin k\right)}{l\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (/ t_m (cos k)))))
   (*
    t_s
    (if (<= t_m 9.8e-164)
      (/ 2.0 (pow (* (/ k (/ l_m (sin k))) t_2) 2.0))
      (if (<= t_m 6e-96)
        (/ 2.0 (/ (* (/ (pow k 2.0) l_m) (/ (* t_m (pow k 2.0)) (cos k))) l_m))
        (if (<= t_m 3.25e-62)
          (/ 2.0 (pow (/ (* t_2 (* k (sin k))) l_m) 2.0))
          (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 = sqrt((t_m / cos(k)));
	double tmp;
	if (t_m <= 9.8e-164) {
		tmp = 2.0 / pow(((k / (l_m / sin(k))) * t_2), 2.0);
	} else if (t_m <= 6e-96) {
		tmp = 2.0 / (((pow(k, 2.0) / l_m) * ((t_m * pow(k, 2.0)) / cos(k))) / l_m);
	} else if (t_m <= 3.25e-62) {
		tmp = 2.0 / pow(((t_2 * (k * sin(k))) / l_m), 2.0);
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 = sqrt((t_m / cos(k)))
    if (t_m <= 9.8d-164) then
        tmp = 2.0d0 / (((k / (l_m / sin(k))) * t_2) ** 2.0d0)
    else if (t_m <= 6d-96) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l_m) * ((t_m * (k ** 2.0d0)) / cos(k))) / l_m)
    else if (t_m <= 3.25d-62) then
        tmp = 2.0d0 / (((t_2 * (k * sin(k))) / l_m) ** 2.0d0)
    else
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-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 = Math.sqrt((t_m / Math.cos(k)));
	double tmp;
	if (t_m <= 9.8e-164) {
		tmp = 2.0 / Math.pow(((k / (l_m / Math.sin(k))) * t_2), 2.0);
	} else if (t_m <= 6e-96) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l_m) * ((t_m * Math.pow(k, 2.0)) / Math.cos(k))) / l_m);
	} else if (t_m <= 3.25e-62) {
		tmp = 2.0 / Math.pow(((t_2 * (k * Math.sin(k))) / l_m), 2.0);
	} else {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -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 = math.sqrt((t_m / math.cos(k)))
	tmp = 0
	if t_m <= 9.8e-164:
		tmp = 2.0 / math.pow(((k / (l_m / math.sin(k))) * t_2), 2.0)
	elif t_m <= 6e-96:
		tmp = 2.0 / (((math.pow(k, 2.0) / l_m) * ((t_m * math.pow(k, 2.0)) / math.cos(k))) / l_m)
	elif t_m <= 3.25e-62:
		tmp = 2.0 / math.pow(((t_2 * (k * math.sin(k))) / l_m), 2.0)
	else:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -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 = sqrt(Float64(t_m / cos(k)))
	tmp = 0.0
	if (t_m <= 9.8e-164)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l_m / sin(k))) * t_2) ^ 2.0));
	elseif (t_m <= 6e-96)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l_m) * Float64(Float64(t_m * (k ^ 2.0)) / cos(k))) / l_m));
	elseif (t_m <= 3.25e-62)
		tmp = Float64(2.0 / (Float64(Float64(t_2 * Float64(k * sin(k))) / l_m) ^ 2.0));
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -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 = sqrt((t_m / cos(k)));
	tmp = 0.0;
	if (t_m <= 9.8e-164)
		tmp = 2.0 / (((k / (l_m / sin(k))) * t_2) ^ 2.0);
	elseif (t_m <= 6e-96)
		tmp = 2.0 / ((((k ^ 2.0) / l_m) * ((t_m * (k ^ 2.0)) / cos(k))) / l_m);
	elseif (t_m <= 3.25e-62)
		tmp = 2.0 / (((t_2 * (k * sin(k))) / l_m) ^ 2.0);
	else
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -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[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.8e-164], N[(2.0 / N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e-96], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.25e-62], N[(2.0 / N[Power[N[(N[(t$95$2 * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 9.7999999999999993e-164

   1. Initial program 46.0%

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

      1. associate-*l* 46.0%

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

      2. sqr-neg 46.0%

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

      3. sqr-neg 46.0%

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

      4. associate-/r* 47.7%

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

      5. distribute-rgt-in 47.7%

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

      6. unpow2 47.7%

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

      7. times-frac 29.7%

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

      8. sqr-neg 29.7%

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

      9. times-frac 47.7%

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

      10. unpow2 47.7%

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

      11. distribute-rgt-in 47.7%

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

   3. Simplified 47.7%

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

      1. add-sqr-sqrt 17.0%

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

      2. pow2 17.0%

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

      3. associate-*l* 16.9%

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

      4. sqrt-prod 8.0%

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

      5. associate-/l/ 7.9%

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

      6. sqrt-div 7.9%

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

      7. sqrt-pow1 9.3%

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

      8. metadata-eval 9.3%

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

      9. sqrt-unprod 7.3%

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

      10. add-sqr-sqrt 10.6%

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

      11. associate-*r* 10.6%

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

   6. Applied egg-rr 10.6%

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

   7. Taylor expanded in k around inf 38.4%

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

      1. associate-/l* 38.4%

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

   9. Simplified 38.4%

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

   if 9.7999999999999993e-164 < t < 6e-96

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. sqr-neg 33.2%

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

      3. sqr-neg 33.2%

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

      4. associate-/r* 42.4%

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

      5. distribute-rgt-in 42.4%

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

      6. unpow2 42.4%

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

      7. times-frac 42.3%

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

      8. sqr-neg 42.3%

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

      9. times-frac 42.4%

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

      10. unpow2 42.4%

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

      11. distribute-rgt-in 42.4%

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

   3. Simplified 42.4%

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

      1. associate-*l* 42.4%

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

      2. associate-*l/ 42.4%

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

      3. associate-*r* 42.4%

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

   6. Applied egg-rr 42.4%

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

   7. Taylor expanded in t around 0 84.3%

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

      1. times-frac 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in k around 0 84.3%

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

   if 6e-96 < t < 3.25000000000000013e-62

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. sqr-neg 33.7%

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

      3. sqr-neg 33.7%

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

      4. associate-/r* 49.7%

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

      5. distribute-rgt-in 49.7%

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

      6. unpow2 49.7%

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

      7. times-frac 49.7%

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

      8. sqr-neg 49.7%

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

      9. times-frac 49.7%

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

      10. unpow2 49.7%

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

      11. distribute-rgt-in 49.7%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

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

      3. associate-*l* 34.2%

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

      4. sqrt-prod 34.2%

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

      5. associate-/l/ 33.3%

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

      6. sqrt-div 37.1%

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

      7. sqrt-pow1 37.1%

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

      8. metadata-eval 37.1%

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

      9. sqrt-unprod 34.5%

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

      10. add-sqr-sqrt 38.0%

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

      11. associate-*r* 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in k around inf 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*r/ 83.5%

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

   9. Simplified 83.5%

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

   if 3.25000000000000013e-62 < t

   1. Initial program 65.9%

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

      1. associate-/r* 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 62.3%

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

      4. sqr-neg 62.3%

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

      5. associate-/r* 62.9%

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

      6. associate-+l+ 62.9%

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

      7. unpow2 62.9%

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

      8. times-frac 52.6%

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

      9. sqr-neg 52.6%

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

      10. times-frac 62.9%

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

      11. unpow2 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. expm1-log1p-u 53.4%

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

      2. expm1-udef 53.3%

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

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

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

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

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

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 71.6%

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

       2. add-sqr-sqrt 32.9%

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

       3. unpow-prod-down 32.9%

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

       4. *-commutative 32.9%

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

       5. sqrt-prod 31.7%

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

       6. sqrt-pow1 31.7%

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

       7. metadata-eval 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 33.4%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 33.4%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {k}^{2}}{\cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt{\frac{t}{\cos k}} \cdot \left(k \cdot \sin k\right)}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\cos k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-174}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{l\_m}{\sin k}} \cdot \sqrt{t\_2}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot t\_2\right) \cdot \frac{\frac{{k}^{2}}{l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (/ t_m (cos k))))
   (*
    t_s
    (if (<= t_m 2.9e-174)
      (/ 2.0 (pow (* (/ k (/ l_m (sin k))) (sqrt t_2)) 2.0))
      (if (<= t_m 4.6e-15)
        (/ 2.0 (* (* (pow (sin k) 2.0) t_2) (/ (/ (pow k 2.0) l_m) l_m)))
        (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 = t_m / cos(k);
	double tmp;
	if (t_m <= 2.9e-174) {
		tmp = 2.0 / pow(((k / (l_m / sin(k))) * sqrt(t_2)), 2.0);
	} else if (t_m <= 4.6e-15) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * t_2) * ((pow(k, 2.0) / l_m) / l_m));
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 = t_m / cos(k)
    if (t_m <= 2.9d-174) then
        tmp = 2.0d0 / (((k / (l_m / sin(k))) * sqrt(t_2)) ** 2.0d0)
    else if (t_m <= 4.6d-15) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * t_2) * (((k ** 2.0d0) / l_m) / l_m))
    else
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-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 = t_m / Math.cos(k);
	double tmp;
	if (t_m <= 2.9e-174) {
		tmp = 2.0 / Math.pow(((k / (l_m / Math.sin(k))) * Math.sqrt(t_2)), 2.0);
	} else if (t_m <= 4.6e-15) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * t_2) * ((Math.pow(k, 2.0) / l_m) / l_m));
	} else {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -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 = t_m / math.cos(k)
	tmp = 0
	if t_m <= 2.9e-174:
		tmp = 2.0 / math.pow(((k / (l_m / math.sin(k))) * math.sqrt(t_2)), 2.0)
	elif t_m <= 4.6e-15:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * t_2) * ((math.pow(k, 2.0) / l_m) / l_m))
	else:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -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(t_m / cos(k))
	tmp = 0.0
	if (t_m <= 2.9e-174)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l_m / sin(k))) * sqrt(t_2)) ^ 2.0));
	elseif (t_m <= 4.6e-15)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * t_2) * Float64(Float64((k ^ 2.0) / l_m) / l_m)));
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -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 = t_m / cos(k);
	tmp = 0.0;
	if (t_m <= 2.9e-174)
		tmp = 2.0 / (((k / (l_m / sin(k))) * sqrt(t_2)) ^ 2.0);
	elseif (t_m <= 4.6e-15)
		tmp = 2.0 / (((sin(k) ^ 2.0) * t_2) * (((k ^ 2.0) / l_m) / l_m));
	else
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -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[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-174], N[(2.0 / N[Power[N[(N[(k / N[(l$95$m / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e-15], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9000000000000001e-174

   1. Initial program 46.6%

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

      1. associate-*l* 46.6%

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

      2. sqr-neg 46.6%

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

      3. sqr-neg 46.6%

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

      4. associate-/r* 48.3%

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

      5. distribute-rgt-in 48.3%

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

      6. unpow2 48.3%

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

      7. times-frac 30.5%

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

      8. sqr-neg 30.5%

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

      9. times-frac 48.3%

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

      10. unpow2 48.3%

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

      11. distribute-rgt-in 48.3%

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

   3. Simplified 48.3%

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

      1. add-sqr-sqrt 16.7%

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

      2. pow2 16.7%

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

      3. associate-*l* 16.7%

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

      4. sqrt-prod 7.5%

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

      5. associate-/l/ 7.5%

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

      6. sqrt-div 7.5%

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

      7. sqrt-pow1 8.9%

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

      8. metadata-eval 8.9%

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

      9. sqrt-unprod 7.5%

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

      10. add-sqr-sqrt 10.2%

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

      11. associate-*r* 10.2%

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

   6. Applied egg-rr 10.2%

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

   7. Taylor expanded in k around inf 37.4%

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

      1. associate-/l* 37.4%

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

   9. Simplified 37.4%

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

   if 2.9000000000000001e-174 < t < 4.59999999999999981e-15

   1. Initial program 40.5%

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

      1. associate-*l* 40.5%

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

      2. sqr-neg 40.5%

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

      3. sqr-neg 40.5%

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

      4. associate-/r* 46.4%

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

      5. distribute-rgt-in 46.4%

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

      6. unpow2 46.4%

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

      7. times-frac 43.6%

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

      8. sqr-neg 43.6%

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

      9. times-frac 46.4%

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

      10. unpow2 46.4%

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

      11. distribute-rgt-in 46.4%

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

   3. Simplified 46.4%

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

      1. associate-*l* 43.7%

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

      2. associate-*l/ 49.8%

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

      3. associate-*r* 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in t around 0 81.5%

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

      1. times-frac 81.4%

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

   9. Simplified 81.4%

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

       1. expm1-log1p-u 66.9%

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

       2. expm1-udef 42.2%

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

       3. associate-/l* 42.2%

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

       4. associate-/l* 42.2%

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

   11. Applied egg-rr 42.2%

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

       1. expm1-def 59.2%

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

       2. expm1-log1p 73.7%

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

       3. associate-/r/ 81.5%

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

       4. associate-/r/ 81.5%

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

   13. Simplified 81.5%

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

   if 4.59999999999999981e-15 < t

   1. Initial program 67.7%

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

      1. associate-/r* 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-*l* 63.5%

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

      4. sqr-neg 63.5%

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

      5. associate-/r* 64.2%

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

      6. associate-+l+ 64.2%

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

      7. unpow2 64.2%

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

      8. times-frac 52.2%

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

      9. sqr-neg 52.2%

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

      10. times-frac 64.2%

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

      11. unpow2 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in k around 0 55.9%

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

      1. expm1-log1p-u 55.5%

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

      2. expm1-udef 55.3%

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

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 74.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 75.4%

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

   9. Simplified 75.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 75.5%

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

   11. Applied egg-rr 75.5%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 75.5%

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

       2. add-sqr-sqrt 35.6%

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

       3. unpow-prod-down 35.6%

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

       4. *-commutative 35.6%

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

       5. sqrt-prod 35.6%

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

       6. sqrt-pow1 35.6%

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

       7. metadata-eval 35.6%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 35.6%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 35.6%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 37.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 37.6%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 37.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 37.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 37.6%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 37.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-174}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{\frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{l\_m}{\frac{{k}^{2}}{l\_m} \cdot \left({\sin k}^{2} \cdot \frac{t\_m}{\cos k}\right)}}}\\ \end{array} \end{array} \]

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 (<= k 1.7e-20)
    (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -2.0)) 2.0)
    (/
     2.0
     (/
      1.0
      (/
       l_m
       (* (/ (pow k 2.0) l_m) (* (pow (sin k) 2.0) (/ t_m (cos 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 (k <= 1.7e-20) {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -2.0)), 2.0);
	} else {
		tmp = 2.0 / (1.0 / (l_m / ((pow(k, 2.0) / l_m) * (pow(sin(k), 2.0) * (t_m / cos(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 (k <= 1.7d-20) then
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-2.0d0))) ** 2.0d0
    else
        tmp = 2.0d0 / (1.0d0 / (l_m / (((k ** 2.0d0) / l_m) * ((sin(k) ** 2.0d0) * (t_m / cos(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 (k <= 1.7e-20) {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -2.0)), 2.0);
	} else {
		tmp = 2.0 / (1.0 / (l_m / ((Math.pow(k, 2.0) / l_m) * (Math.pow(Math.sin(k), 2.0) * (t_m / Math.cos(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 k <= 1.7e-20:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -2.0)), 2.0)
	else:
		tmp = 2.0 / (1.0 / (l_m / ((math.pow(k, 2.0) / l_m) * (math.pow(math.sin(k), 2.0) * (t_m / math.cos(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 (k <= 1.7e-20)
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(1.0 / Float64(l_m / Float64(Float64((k ^ 2.0) / l_m) * Float64((sin(k) ^ 2.0) * Float64(t_m / cos(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 (k <= 1.7e-20)
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	else
		tmp = 2.0 / (1.0 / (l_m / (((k ^ 2.0) / l_m) * ((sin(k) ^ 2.0) * (t_m / cos(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[k, 1.7e-20], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(1.0 / N[(l$95$m / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.6999999999999999e-20

   1. Initial program 53.2%

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

      1. associate-/r* 53.2%

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

      2. sqr-neg 53.2%

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

      3. associate-*l* 49.8%

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

      4. sqr-neg 49.8%

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

      5. associate-/r* 51.8%

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

      6. associate-+l+ 51.8%

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

      7. unpow2 51.8%

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

      8. times-frac 36.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 36.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 51.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 51.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 35.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 35.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 33.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 35.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 35.7%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 35.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 35.7%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 35.7%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 35.7%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{-1}}\right)}^{2} \]

       2. add-sqr-sqrt 15.7%

          \[\leadsto {\left(\ell \cdot {\color{blue}{\left(\sqrt{k \cdot {t}^{1.5}} \cdot \sqrt{k \cdot {t}^{1.5}}\right)}}^{-1}\right)}^{2} \]

       3. unpow-prod-down 15.7%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)}\right)}^{2} \]

       4. *-commutative 15.7%

          \[\leadsto {\left(\ell \cdot \left({\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       5. sqrt-prod 11.3%

          \[\leadsto {\left(\ell \cdot \left({\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       6. sqrt-pow1 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       7. metadata-eval 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 12.1%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 12.1%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 12.1%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 12.1%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 12.1%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 12.1%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]

   if 1.6999999999999999e-20 < k

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 48.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 48.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 48.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 48.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 48.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 36.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 36.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 48.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 48.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 48.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 48.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 50.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 50.1%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 50.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 73.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 73.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 73.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]
   10. Step-by-step derivation

       1. clear-num 73.0%

          \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}}} \]

       2. inv-pow 73.0%

          \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\right)}^{-1}}} \]

       3. associate-/l* 73.0%

          \[\leadsto \frac{2}{{\left(\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}}\right)}^{-1}} \]

   11. Applied egg-rr 73.0%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)}^{-1}}} \]
   12. Step-by-step derivation

       1. unpow-1 73.0%

          \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}}}} \]

       2. associate-/r/ 73.1%

          \[\leadsto \frac{2}{\frac{1}{\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}}}} \]

   13. Simplified 73.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\frac{{k}^{2}}{\ell} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{l\_m}}{l\_m}}\\ \end{array} \end{array} \]

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 (<= k 1.45e-15)
    (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -2.0)) 2.0)
    (/
     2.0
     (/ (* (/ (* t_m (pow k 2.0)) (cos k)) (/ (pow (sin k) 2.0) l_m)) 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 (k <= 1.45e-15) {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -2.0)), 2.0);
	} else {
		tmp = 2.0 / ((((t_m * pow(k, 2.0)) / cos(k)) * (pow(sin(k), 2.0) / l_m)) / l_m);
	}
	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 (k <= 1.45d-15) then
        tmp = (l_m * (((t_m ** 0.75d0) * sqrt(k)) ** (-2.0d0))) ** 2.0d0
    else
        tmp = 2.0d0 / ((((t_m * (k ** 2.0d0)) / cos(k)) * ((sin(k) ** 2.0d0) / l_m)) / l_m)
    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 (k <= 1.45e-15) {
		tmp = Math.pow((l_m * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -2.0)), 2.0);
	} else {
		tmp = 2.0 / ((((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / l_m)) / l_m);
	}
	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 k <= 1.45e-15:
		tmp = math.pow((l_m * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -2.0)), 2.0)
	else:
		tmp = 2.0 / ((((t_m * math.pow(k, 2.0)) / math.cos(k)) * (math.pow(math.sin(k), 2.0) / l_m)) / 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 (k <= 1.45e-15)
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * Float64((sin(k) ^ 2.0) / l_m)) / l_m));
	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 (k <= 1.45e-15)
		tmp = (l_m * (((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	else
		tmp = 2.0 / ((((t_m * (k ^ 2.0)) / cos(k)) * ((sin(k) ^ 2.0) / l_m)) / l_m);
	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[k, 1.45e-15], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45000000000000009e-15

   1. Initial program 53.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 53.2%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 49.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 49.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 51.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 51.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 51.8%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 36.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 36.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 51.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 51.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 35.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 35.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 33.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 35.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 35.7%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 35.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 35.7%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 35.7%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 35.7%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{-1}}\right)}^{2} \]

       2. add-sqr-sqrt 15.7%

          \[\leadsto {\left(\ell \cdot {\color{blue}{\left(\sqrt{k \cdot {t}^{1.5}} \cdot \sqrt{k \cdot {t}^{1.5}}\right)}}^{-1}\right)}^{2} \]

       3. unpow-prod-down 15.7%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)}\right)}^{2} \]

       4. *-commutative 15.7%

          \[\leadsto {\left(\ell \cdot \left({\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       5. sqrt-prod 11.3%

          \[\leadsto {\left(\ell \cdot \left({\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       6. sqrt-pow1 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       7. metadata-eval 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 11.3%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 12.1%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 12.1%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 12.1%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 12.1%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 12.1%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 12.1%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]

   if 1.45000000000000009e-15 < k

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 48.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 48.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 48.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 48.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 48.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 36.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 36.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 48.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 48.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 48.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 48.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 50.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 50.1%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 50.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 73.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r* 73.0%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

      2. *-commutative 73.0%

         \[\leadsto \frac{2}{\frac{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot \ell}}}{\ell}} \]

      3. times-frac 73.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

   9. Simplified 73.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t\_m \cdot 0.16666666666666666}{l\_m}, \frac{{k}^{4}}{\frac{l\_m}{t\_m}}\right)}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \]

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 1.15e-62)
    (/
     2.0
     (/
      (fma
       (pow k 6.0)
       (/ (* t_m 0.16666666666666666) l_m)
       (/ (pow k 4.0) (/ l_m t_m)))
      l_m))
    (pow (* l_m (pow (* (pow t_m 0.75) (sqrt k)) -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 <= 1.15e-62) {
		tmp = 2.0 / (fma(pow(k, 6.0), ((t_m * 0.16666666666666666) / l_m), (pow(k, 4.0) / (l_m / t_m))) / l_m);
	} else {
		tmp = pow((l_m * pow((pow(t_m, 0.75) * sqrt(k)), -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 <= 1.15e-62)
		tmp = Float64(2.0 / Float64(fma((k ^ 6.0), Float64(Float64(t_m * 0.16666666666666666) / l_m), Float64((k ^ 4.0) / Float64(l_m / t_m))) / l_m));
	else
		tmp = Float64(l_m * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.15e-62], N[(2.0 / N[(N[(N[Power[k, 6.0], $MachinePrecision] * N[(N[(t$95$m * 0.16666666666666666), $MachinePrecision] / l$95$m), $MachinePrecision] + N[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.15e-62

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.6%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 44.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{6} \cdot \left(-0.3333333333333333 \cdot \frac{t}{\ell} - -0.5 \cdot \frac{t}{\ell}\right) + \frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. fma-def 44.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left({k}^{6}, -0.3333333333333333 \cdot \frac{t}{\ell} - -0.5 \cdot \frac{t}{\ell}, \frac{{k}^{4} \cdot t}{\ell}\right)}}{\ell}} \]

       2. distribute-rgt-out-- 54.2%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \color{blue}{\frac{t}{\ell} \cdot \left(-0.3333333333333333 - -0.5\right)}, \frac{{k}^{4} \cdot t}{\ell}\right)}{\ell}} \]

       3. metadata-eval 54.2%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t}{\ell} \cdot \color{blue}{0.16666666666666666}, \frac{{k}^{4} \cdot t}{\ell}\right)}{\ell}} \]

       4. associate-*l/ 54.2%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \color{blue}{\frac{t \cdot 0.16666666666666666}{\ell}}, \frac{{k}^{4} \cdot t}{\ell}\right)}{\ell}} \]

       5. associate-/l* 56.9%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t \cdot 0.16666666666666666}{\ell}, \color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}\right)}{\ell}} \]

   12. Simplified 56.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left({k}^{6}, \frac{t \cdot 0.16666666666666666}{\ell}, \frac{{k}^{4}}{\frac{\ell}{t}}\right)}}{\ell}} \]

   if 1.15e-62 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 62.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 62.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 53.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 71.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{-1}}\right)}^{2} \]

       2. add-sqr-sqrt 32.9%

          \[\leadsto {\left(\ell \cdot {\color{blue}{\left(\sqrt{k \cdot {t}^{1.5}} \cdot \sqrt{k \cdot {t}^{1.5}}\right)}}^{-1}\right)}^{2} \]

       3. unpow-prod-down 32.9%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)}\right)}^{2} \]

       4. *-commutative 32.9%

          \[\leadsto {\left(\ell \cdot \left({\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       5. sqrt-prod 31.7%

          \[\leadsto {\left(\ell \cdot \left({\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       6. sqrt-pow1 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       7. metadata-eval 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 31.7%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

       11. metadata-eval 33.4%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 33.4%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 33.4%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 33.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t \cdot 0.16666666666666666}{\ell}, \frac{{k}^{4}}{\frac{\ell}{t}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t\_m \cdot 0.16666666666666666}{l\_m}, \frac{{k}^{4}}{\frac{l\_m}{t\_m}}\right)}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left({\left({t\_m}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)\right)}^{2}\\ \end{array} \end{array} \]

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 1.2e-62)
    (/
     2.0
     (/
      (fma
       (pow k 6.0)
       (/ (* t_m 0.16666666666666666) l_m)
       (/ (pow k 4.0) (/ l_m t_m)))
      l_m))
    (pow (* l_m (* (pow (pow t_m 1.5) -1.0) (/ 1.0 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 tmp;
	if (t_m <= 1.2e-62) {
		tmp = 2.0 / (fma(pow(k, 6.0), ((t_m * 0.16666666666666666) / l_m), (pow(k, 4.0) / (l_m / t_m))) / l_m);
	} else {
		tmp = pow((l_m * (pow(pow(t_m, 1.5), -1.0) * (1.0 / 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)
	tmp = 0.0
	if (t_m <= 1.2e-62)
		tmp = Float64(2.0 / Float64(fma((k ^ 6.0), Float64(Float64(t_m * 0.16666666666666666) / l_m), Float64((k ^ 4.0) / Float64(l_m / t_m))) / l_m));
	else
		tmp = Float64(l_m * Float64(((t_m ^ 1.5) ^ -1.0) * Float64(1.0 / k))) ^ 2.0;
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-62], N[(2.0 / N[(N[(N[Power[k, 6.0], $MachinePrecision] * N[(N[(t$95$m * 0.16666666666666666), $MachinePrecision] / l$95$m), $MachinePrecision] + N[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[Power[N[Power[t$95$m, 1.5], $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.19999999999999992e-62

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.6%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 44.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{6} \cdot \left(-0.3333333333333333 \cdot \frac{t}{\ell} - -0.5 \cdot \frac{t}{\ell}\right) + \frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. fma-def 44.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left({k}^{6}, -0.3333333333333333 \cdot \frac{t}{\ell} - -0.5 \cdot \frac{t}{\ell}, \frac{{k}^{4} \cdot t}{\ell}\right)}}{\ell}} \]

       2. distribute-rgt-out-- 54.2%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \color{blue}{\frac{t}{\ell} \cdot \left(-0.3333333333333333 - -0.5\right)}, \frac{{k}^{4} \cdot t}{\ell}\right)}{\ell}} \]

       3. metadata-eval 54.2%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t}{\ell} \cdot \color{blue}{0.16666666666666666}, \frac{{k}^{4} \cdot t}{\ell}\right)}{\ell}} \]

       4. associate-*l/ 54.2%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \color{blue}{\frac{t \cdot 0.16666666666666666}{\ell}}, \frac{{k}^{4} \cdot t}{\ell}\right)}{\ell}} \]

       5. associate-/l* 56.9%

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t \cdot 0.16666666666666666}{\ell}, \color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}\right)}{\ell}} \]

   12. Simplified 56.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left({k}^{6}, \frac{t \cdot 0.16666666666666666}{\ell}, \frac{{k}^{4}}{\frac{\ell}{t}}\right)}}{\ell}} \]

   if 1.19999999999999992e-62 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 62.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 62.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 53.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 71.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{-1}}\right)}^{2} \]

       2. *-commutative 71.6%

          \[\leadsto {\left(\ell \cdot {\color{blue}{\left({t}^{1.5} \cdot k\right)}}^{-1}\right)}^{2} \]

       3. unpow-prod-down 71.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{-1} \cdot {k}^{-1}\right)}\right)}^{2} \]

       4. inv-pow 71.6%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{1.5}\right)}^{-1} \cdot \color{blue}{\frac{1}{k}}\right)\right)}^{2} \]

   13. Applied egg-rr 71.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({k}^{6}, \frac{t \cdot 0.16666666666666666}{\ell}, \frac{{k}^{4}}{\frac{\ell}{t}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left({\left({t}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{l\_m} \cdot \frac{t\_m \cdot {k}^{2}}{\cos k}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left({\left({t\_m}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)\right)}^{2}\\ \end{array} \end{array} \]

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 1.45e-58)
    (/ 2.0 (/ (* (/ (pow k 2.0) l_m) (/ (* t_m (pow k 2.0)) (cos k))) l_m))
    (pow (* l_m (* (pow (pow t_m 1.5) -1.0) (/ 1.0 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 tmp;
	if (t_m <= 1.45e-58) {
		tmp = 2.0 / (((pow(k, 2.0) / l_m) * ((t_m * pow(k, 2.0)) / cos(k))) / l_m);
	} else {
		tmp = pow((l_m * (pow(pow(t_m, 1.5), -1.0) * (1.0 / k))), 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 <= 1.45d-58) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l_m) * ((t_m * (k ** 2.0d0)) / cos(k))) / l_m)
    else
        tmp = (l_m * (((t_m ** 1.5d0) ** (-1.0d0)) * (1.0d0 / k))) ** 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 <= 1.45e-58) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l_m) * ((t_m * Math.pow(k, 2.0)) / Math.cos(k))) / l_m);
	} else {
		tmp = Math.pow((l_m * (Math.pow(Math.pow(t_m, 1.5), -1.0) * (1.0 / 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):
	tmp = 0
	if t_m <= 1.45e-58:
		tmp = 2.0 / (((math.pow(k, 2.0) / l_m) * ((t_m * math.pow(k, 2.0)) / math.cos(k))) / l_m)
	else:
		tmp = math.pow((l_m * (math.pow(math.pow(t_m, 1.5), -1.0) * (1.0 / 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)
	tmp = 0.0
	if (t_m <= 1.45e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l_m) * Float64(Float64(t_m * (k ^ 2.0)) / cos(k))) / l_m));
	else
		tmp = Float64(l_m * Float64(((t_m ^ 1.5) ^ -1.0) * Float64(1.0 / 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)
	tmp = 0.0;
	if (t_m <= 1.45e-58)
		tmp = 2.0 / ((((k ^ 2.0) / l_m) * ((t_m * (k ^ 2.0)) / cos(k))) / l_m);
	else
		tmp = (l_m * (((t_m ^ 1.5) ^ -1.0) * (1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-58], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[Power[N[Power[t$95$m, 1.5], $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999995e-58

   1. Initial program 45.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 45.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 45.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 45.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.7%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.9%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 66.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 56.4%

       \[\leadsto \frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\cos k}}{\ell}} \]

   if 1.44999999999999995e-58 < t

   1. Initial program 65.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.5%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 61.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 61.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.5%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.5%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 54.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 54.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 71.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 72.4%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 72.4%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 72.4%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 72.4%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{-1}}\right)}^{2} \]

       2. *-commutative 72.4%

          \[\leadsto {\left(\ell \cdot {\color{blue}{\left({t}^{1.5} \cdot k\right)}}^{-1}\right)}^{2} \]

       3. unpow-prod-down 72.4%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{-1} \cdot {k}^{-1}\right)}\right)}^{2} \]

       4. inv-pow 72.4%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{1.5}\right)}^{-1} \cdot \color{blue}{\frac{1}{k}}\right)\right)}^{2} \]

   13. Applied egg-rr 72.4%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left({\left({t}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{l\_m}{t\_m}}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \left({\left({t\_m}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)\right)}^{2}\\ \end{array} \end{array} \]

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 9.5e-63)
    (/ 2.0 (/ (/ (pow k 4.0) (/ l_m t_m)) l_m))
    (pow (* l_m (* (pow (pow t_m 1.5) -1.0) (/ 1.0 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 tmp;
	if (t_m <= 9.5e-63) {
		tmp = 2.0 / ((pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = pow((l_m * (pow(pow(t_m, 1.5), -1.0) * (1.0 / k))), 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 <= 9.5d-63) then
        tmp = 2.0d0 / (((k ** 4.0d0) / (l_m / t_m)) / l_m)
    else
        tmp = (l_m * (((t_m ** 1.5d0) ** (-1.0d0)) * (1.0d0 / k))) ** 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 <= 9.5e-63) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = Math.pow((l_m * (Math.pow(Math.pow(t_m, 1.5), -1.0) * (1.0 / 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):
	tmp = 0
	if t_m <= 9.5e-63:
		tmp = 2.0 / ((math.pow(k, 4.0) / (l_m / t_m)) / l_m)
	else:
		tmp = math.pow((l_m * (math.pow(math.pow(t_m, 1.5), -1.0) * (1.0 / 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)
	tmp = 0.0
	if (t_m <= 9.5e-63)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l_m / t_m)) / l_m));
	else
		tmp = Float64(l_m * Float64(((t_m ^ 1.5) ^ -1.0) * Float64(1.0 / 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)
	tmp = 0.0;
	if (t_m <= 9.5e-63)
		tmp = 2.0 / (((k ^ 4.0) / (l_m / t_m)) / l_m);
	else
		tmp = (l_m * (((t_m ^ 1.5) ^ -1.0) * (1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 9.5e-63], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[Power[N[Power[t$95$m, 1.5], $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.50000000000000016e-63

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.6%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 53.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. associate-/l* 56.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   12. Simplified 56.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   if 9.50000000000000016e-63 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 62.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 62.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 53.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. inv-pow 71.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{-1}}\right)}^{2} \]

       2. *-commutative 71.6%

          \[\leadsto {\left(\ell \cdot {\color{blue}{\left({t}^{1.5} \cdot k\right)}}^{-1}\right)}^{2} \]

       3. unpow-prod-down 71.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{-1} \cdot {k}^{-1}\right)}\right)}^{2} \]

       4. inv-pow 71.6%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{1.5}\right)}^{-1} \cdot \color{blue}{\frac{1}{k}}\right)\right)}^{2} \]

   13. Applied egg-rr 71.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left({\left({t}^{1.5}\right)}^{-1} \cdot \frac{1}{k}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{l\_m}{t\_m}}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \frac{1}{k \cdot {t\_m}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

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 1.1e-62)
    (/ 2.0 (/ (/ (pow k 4.0) (/ l_m t_m)) l_m))
    (pow (* l_m (/ 1.0 (* k (pow t_m 1.5)))) 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 <= 1.1e-62) {
		tmp = 2.0 / ((pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = pow((l_m * (1.0 / (k * pow(t_m, 1.5)))), 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 <= 1.1d-62) then
        tmp = 2.0d0 / (((k ** 4.0d0) / (l_m / t_m)) / l_m)
    else
        tmp = (l_m * (1.0d0 / (k * (t_m ** 1.5d0)))) ** 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 <= 1.1e-62) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = Math.pow((l_m * (1.0 / (k * Math.pow(t_m, 1.5)))), 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 <= 1.1e-62:
		tmp = 2.0 / ((math.pow(k, 4.0) / (l_m / t_m)) / l_m)
	else:
		tmp = math.pow((l_m * (1.0 / (k * math.pow(t_m, 1.5)))), 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 <= 1.1e-62)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l_m / t_m)) / l_m));
	else
		tmp = Float64(l_m * Float64(1.0 / Float64(k * (t_m ^ 1.5)))) ^ 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 <= 1.1e-62)
		tmp = 2.0 / (((k ^ 4.0) / (l_m / t_m)) / l_m);
	else
		tmp = (l_m * (1.0 / (k * (t_m ^ 1.5)))) ^ 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, 1.1e-62], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(1.0 / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.10000000000000009e-62

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.6%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 53.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. associate-/l* 56.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   12. Simplified 56.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   if 1.10000000000000009e-62 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 62.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 62.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 53.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{l\_m}{t\_m}}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(l\_m \cdot \frac{\frac{1}{k}}{{t\_m}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

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 1.1e-62)
    (/ 2.0 (/ (/ (pow k 4.0) (/ l_m t_m)) l_m))
    (pow (* l_m (/ (/ 1.0 k) (pow t_m 1.5))) 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 <= 1.1e-62) {
		tmp = 2.0 / ((pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = pow((l_m * ((1.0 / k) / pow(t_m, 1.5))), 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 <= 1.1d-62) then
        tmp = 2.0d0 / (((k ** 4.0d0) / (l_m / t_m)) / l_m)
    else
        tmp = (l_m * ((1.0d0 / k) / (t_m ** 1.5d0))) ** 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 <= 1.1e-62) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = Math.pow((l_m * ((1.0 / k) / Math.pow(t_m, 1.5))), 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 <= 1.1e-62:
		tmp = 2.0 / ((math.pow(k, 4.0) / (l_m / t_m)) / l_m)
	else:
		tmp = math.pow((l_m * ((1.0 / k) / math.pow(t_m, 1.5))), 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 <= 1.1e-62)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l_m / t_m)) / l_m));
	else
		tmp = Float64(l_m * Float64(Float64(1.0 / k) / (t_m ^ 1.5))) ^ 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 <= 1.1e-62)
		tmp = 2.0 / (((k ^ 4.0) / (l_m / t_m)) / l_m);
	else
		tmp = (l_m * ((1.0 / k) / (t_m ^ 1.5))) ^ 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, 1.1e-62], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m * N[(N[(1.0 / k), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.10000000000000009e-62

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.6%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 53.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. associate-/l* 56.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   12. Simplified 56.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   if 1.10000000000000009e-62 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 62.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 62.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 53.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 71.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 71.6%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 58.8%

          \[\leadsto {\left(\ell \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{k \cdot {t}^{1.5}}\right)\right)}\right)}^{2} \]

       2. expm1-udef 49.8%

          \[\leadsto {\left(\ell \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{k \cdot {t}^{1.5}}\right)} - 1\right)}\right)}^{2} \]

       3. associate-/r* 49.8%

          \[\leadsto {\left(\ell \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{k}}{{t}^{1.5}}}\right)} - 1\right)\right)}^{2} \]

   13. Applied egg-rr 49.8%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{1.5}}\right)} - 1\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. expm1-def 58.8%

          \[\leadsto {\left(\ell \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{1.5}}\right)\right)}\right)}^{2} \]

       2. expm1-log1p 71.6%

          \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{1}{k}}{{t}^{1.5}}}\right)}^{2} \]

   15. Simplified 71.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\frac{\frac{1}{k}}{{t}^{1.5}}}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{1}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 72.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{l\_m}{t\_m}}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k \cdot {t\_m}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

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 1.15e-62)
    (/ 2.0 (/ (/ (pow k 4.0) (/ l_m t_m)) l_m))
    (pow (/ l_m (* k (pow t_m 1.5))) 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 <= 1.15e-62) {
		tmp = 2.0 / ((pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = pow((l_m / (k * pow(t_m, 1.5))), 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 <= 1.15d-62) then
        tmp = 2.0d0 / (((k ** 4.0d0) / (l_m / t_m)) / l_m)
    else
        tmp = (l_m / (k * (t_m ** 1.5d0))) ** 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 <= 1.15e-62) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / (l_m / t_m)) / l_m);
	} else {
		tmp = Math.pow((l_m / (k * Math.pow(t_m, 1.5))), 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 <= 1.15e-62:
		tmp = 2.0 / ((math.pow(k, 4.0) / (l_m / t_m)) / l_m)
	else:
		tmp = math.pow((l_m / (k * math.pow(t_m, 1.5))), 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 <= 1.15e-62)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l_m / t_m)) / l_m));
	else
		tmp = Float64(l_m / Float64(k * (t_m ^ 1.5))) ^ 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 <= 1.15e-62)
		tmp = 2.0 / (((k ^ 4.0) / (l_m / t_m)) / l_m);
	else
		tmp = (l_m / (k * (t_m ^ 1.5))) ^ 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, 1.15e-62], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(l$95$m / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.15e-62

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 44.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 47.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 31.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 47.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 47.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. associate-*l/ 46.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

      3. associate-*r* 46.6%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. times-frac 67.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

   10. Taylor expanded in k around 0 53.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. associate-/l* 56.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   12. Simplified 56.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   if 1.15e-62 < t

   1. Initial program 65.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 65.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 62.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 62.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 62.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 62.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 52.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 62.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 53.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 53.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 71.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\frac{t\_m \cdot {k}^{4}}{l\_m}}{l\_m}} \end{array} \]

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 (/ (/ (* t_m (pow k 4.0)) l_m) 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 / (((t_m * pow(k, 4.0)) / l_m) / 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 / (((t_m * (k ** 4.0d0)) / l_m) / 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 / (((t_m * Math.pow(k, 4.0)) / l_m) / 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 / (((t_m * math.pow(k, 4.0)) / l_m) / 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(Float64(t_m * (k ^ 4.0)) / l_m) / 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 / (((t_m * (k ^ 4.0)) / l_m) / 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[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 51.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. sqr-neg 51.9%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. sqr-neg 51.9%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 53.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 53.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 53.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 39.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 39.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 53.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 53.9%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 53.9%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 51.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   2. associate-*l/ 52.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

   3. associate-*r* 52.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

6. Applied egg-rr 52.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

7. Taylor expanded in t around 0 63.6%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
8. Step-by-step derivation

   1. times-frac 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

9. Simplified 64.5%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

10. Taylor expanded in k around 0 52.3%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]

11. Final simplification 52.3%

    \[\leadsto \frac{2}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}} \]
12. Add Preprocessing

Alternative 21: 56.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\frac{{k}^{4}}{\frac{l\_m}{t\_m}}}{l\_m}} \end{array} \]

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 (/ (/ (pow k 4.0) (/ l_m t_m)) 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 / ((pow(k, 4.0) / (l_m / t_m)) / 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 / (((k ** 4.0d0) / (l_m / t_m)) / 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 / ((Math.pow(k, 4.0) / (l_m / t_m)) / 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 / ((math.pow(k, 4.0) / (l_m / t_m)) / 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((k ^ 4.0) / Float64(l_m / t_m)) / 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 / (((k ^ 4.0) / (l_m / t_m)) / 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[(N[Power[k, 4.0], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 51.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. sqr-neg 51.9%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. sqr-neg 51.9%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 53.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 53.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 53.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 39.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 39.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 53.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 53.9%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 53.9%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 51.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   2. associate-*l/ 52.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]

   3. associate-*r* 52.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

6. Applied egg-rr 52.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

7. Taylor expanded in t around 0 63.6%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
8. Step-by-step derivation

   1. times-frac 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

9. Simplified 64.5%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}{\ell}} \]

10. Taylor expanded in k around 0 52.3%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
11. Step-by-step derivation

    1. associate-/l* 54.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

12. Simplified 54.0%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

13. Final simplification 54.0%

    \[\leadsto \frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))