Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 85.6%

Time: 23.1s

Alternatives: 22

Speedup: 1.9×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 9 \cdot 10^{-194}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-194)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (pow
      (*
       (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-194) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / pow((((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-194) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow((((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-194)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-194], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $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 < 8.9999999999999997e-194

   1. Initial program 50.1%

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

      1. associate-*l* 50.1%

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

      2. sqr-neg 50.1%

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

      3. sqr-neg 50.1%

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

      4. associate-/r* 58.4%

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

      5. distribute-rgt-in 58.4%

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

      6. unpow2 58.4%

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

      7. times-frac 38.9%

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

      8. sqr-neg 38.9%

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

      9. times-frac 58.4%

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

      10. unpow2 58.4%

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

      11. distribute-rgt-in 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in t around 0 61.2%

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

      1. associate-*r* 61.2%

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

      2. times-frac 63.8%

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

   7. Simplified 63.8%

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

   if 8.9999999999999997e-194 < t

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. sqr-neg 53.6%

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

      3. sqr-neg 53.6%

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

      4. associate-/r* 59.8%

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

      5. distribute-rgt-in 59.8%

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

      6. unpow2 59.8%

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

      7. times-frac 51.0%

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

      8. sqr-neg 51.0%

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

      9. times-frac 59.8%

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

      10. unpow2 59.8%

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

      11. distribute-rgt-in 59.8%

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

   3. Simplified 59.8%

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

      1. add-cube-cbrt 59.7%

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

      2. pow3 59.7%

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

      3. *-commutative 59.7%

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

      4. cbrt-prod 59.6%

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

      5. associate-/l/ 53.4%

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

      6. cbrt-div 53.4%

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

      7. rem-cbrt-cube 63.1%

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

      8. cbrt-unprod 82.3%

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

      9. pow2 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*r/ 82.3%

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

   8. Applied egg-rr 82.3%

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

      1. add-cube-cbrt 82.1%

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

      2. pow3 82.2%

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

      3. div-inv 82.2%

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

      4. pow-flip 82.2%

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

      5. metadata-eval 82.2%

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

   10. Applied egg-rr 82.2%

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

       1. add-cube-cbrt 82.1%

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

       2. pow3 82.1%

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

   12. Applied egg-rr 87.7%

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

       1. *-commutative 87.7%

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

   14. Simplified 87.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-194}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t_2 + 1\right)\right) \leq 10^{+298}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + t_2\right)\right) \cdot \left(\sin k \cdot \left(\frac{t_m}{\ell} \cdot \frac{{t_m}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))
          (+ 1.0 (+ t_2 1.0)))
         1e+298)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 t_2))
        (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l)))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))) <= 1e+298) {
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))) <= 1e+298) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_2 + 1.0))) <= 1e+298)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l)))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+298], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 9.9999999999999996e297

   1. Initial program 85.2%

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

      1. associate-*l* 85.2%

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

      2. sqr-neg 85.2%

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

      3. sqr-neg 85.2%

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

      4. associate-/r* 88.5%

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

      5. distribute-rgt-in 88.5%

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

      6. unpow2 88.5%

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

      7. times-frac 66.1%

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

      8. sqr-neg 66.1%

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

      9. times-frac 88.5%

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

      10. unpow2 88.5%

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

      11. distribute-rgt-in 88.5%

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

   3. Simplified 88.5%

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

      1. associate-/l/ 85.2%

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

      2. unpow3 85.2%

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

      3. times-frac 92.6%

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

      4. pow2 92.6%

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

   6. Applied egg-rr 92.6%

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

   if 9.9999999999999996e297 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 23.6%

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

      1. associate-/r* 23.6%

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

      2. sqr-neg 23.6%

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

      3. associate-*l* 20.7%

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

      4. sqr-neg 20.7%

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

      5. associate-/r* 31.4%

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

      6. associate-+l+ 31.4%

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

      7. unpow2 31.4%

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

      8. times-frac 22.5%

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

      9. sqr-neg 22.5%

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

      10. times-frac 31.4%

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

      11. unpow2 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in k around 0 28.2%

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

      1. add-cube-cbrt 28.2%

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

      2. pow3 28.2%

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

      3. associate-/r* 26.6%

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

      4. cbrt-div 26.6%

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

      5. rem-cbrt-cube 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. pow2 37.6%

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

      2. cbrt-div 40.9%

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

      3. unpow2 40.9%

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

      4. cbrt-prod 51.4%

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

      5. unpow2 51.4%

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

      6. cbrt-prod 55.5%

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

      7. pow2 55.5%

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

   9. Applied egg-rr 55.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \leq 10^{+298}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if ((math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))) <= math.inf:
		tmp = 2.0 / ((math.tan(k) * (2.0 + t_2)) * (math.sin(k) * ((t_m / l) * (math.pow(t_m, 2.0) / l))))
	else:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (math.sin(k) * math.tan(k))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_2 + 1.0))) <= Inf)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(sin(k) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (((tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))) <= Inf)
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m / l) * ((t_m ^ 2.0) / l))));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (sin(k) * tan(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.6%

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

      1. associate-*l* 84.6%

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

      2. sqr-neg 84.6%

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

      3. sqr-neg 84.6%

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

      4. associate-/r* 87.0%

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

      5. distribute-rgt-in 87.0%

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

      6. unpow2 87.0%

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

      7. times-frac 65.9%

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

      8. sqr-neg 65.9%

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

      9. times-frac 87.0%

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

      10. unpow2 87.0%

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

      11. distribute-rgt-in 87.0%

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

   3. Simplified 87.0%

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

      1. associate-/l/ 84.6%

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

      2. unpow3 84.6%

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

      3. times-frac 90.7%

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

      4. pow2 90.7%

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

   6. Applied egg-rr 90.7%

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

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

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. sqr-neg 0.0%

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

      3. sqr-neg 0.0%

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

      4. associate-/r* 15.2%

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

      5. distribute-rgt-in 15.2%

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

      6. unpow2 15.2%

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

      7. times-frac 9.7%

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

      8. sqr-neg 9.7%

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

      9. times-frac 15.2%

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

      10. unpow2 15.2%

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

      11. distribute-rgt-in 15.2%

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

   3. Simplified 15.2%

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

      1. add-sqr-sqrt 5.8%

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

      2. pow2 5.8%

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

      3. associate-/l/ 0.0%

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

      4. sqrt-div 0.0%

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

      5. sqrt-pow1 6.5%

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

      6. metadata-eval 6.5%

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

      7. sqrt-unprod 10.1%

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

      8. add-sqr-sqrt 23.2%

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

   6. Applied egg-rr 23.2%

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

   7. Taylor expanded in t around inf 21.5%

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

      1. expm1-log1p-u 20.0%

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

      2. expm1-udef 14.0%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)\right)} - 1}} \]

      3. associate-*l* 14.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)\right)}\right)} - 1} \]

      4. tan-quot 14.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(2 \cdot \color{blue}{\tan k}\right)\right)\right)} - 1} \]

      5. *-commutative 14.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 2\right)}\right)\right)} - 1} \]

   9. Applied egg-rr 14.0%

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

       1. expm1-def 20.0%

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

       2. expm1-log1p 21.4%

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

       3. associate-*r* 21.4%

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

       4. *-commutative 21.4%

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

   11. Simplified 21.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \leq \infty:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-142)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-142) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-142) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} 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), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-142)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	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) ^ 2.0))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-142], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.10000000000000008e-142

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 1.10000000000000008e-142 < t

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. sqr-neg 55.9%

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

      3. sqr-neg 55.9%

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

      4. associate-/r* 62.1%

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

      5. distribute-rgt-in 62.1%

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

      6. unpow2 62.1%

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

      7. times-frac 56.1%

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

      8. sqr-neg 56.1%

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

      9. times-frac 62.1%

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

      10. unpow2 62.1%

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

      11. distribute-rgt-in 62.1%

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

   3. Simplified 62.1%

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

      1. add-cube-cbrt 62.0%

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

      2. pow3 62.0%

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

      3. *-commutative 62.0%

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

      4. cbrt-prod 61.9%

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

      5. associate-/l/ 55.7%

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

      6. cbrt-div 55.7%

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

      7. rem-cbrt-cube 61.6%

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

      8. cbrt-unprod 81.3%

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

      9. pow2 81.3%

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

   6. Applied egg-rr 81.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot {\left(\frac{t_m \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-142)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (/ (* t_m (cbrt (sin k))) (pow (cbrt l) 2.0)) 3.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-142) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow(((t_m * cbrt(sin(k))) / pow(cbrt(l), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-142) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow(((t_m * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e-142)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(Float64(t_m * cbrt(sin(k))) / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-142], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2500000000000001e-142

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 1.2500000000000001e-142 < t

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. sqr-neg 55.9%

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

      3. sqr-neg 55.9%

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

      4. associate-/r* 62.1%

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

      5. distribute-rgt-in 62.1%

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

      6. unpow2 62.1%

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

      7. times-frac 56.1%

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

      8. sqr-neg 56.1%

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

      9. times-frac 62.1%

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

      10. unpow2 62.1%

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

      11. distribute-rgt-in 62.1%

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

   3. Simplified 62.1%

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

      1. add-cube-cbrt 62.0%

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

      2. pow3 62.0%

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

      3. *-commutative 62.0%

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

      4. cbrt-prod 61.9%

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

      5. associate-/l/ 55.7%

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

      6. cbrt-div 55.7%

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

      7. rem-cbrt-cube 61.6%

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

      8. cbrt-unprod 81.3%

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

      9. pow2 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. associate-*r/ 81.2%

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

   8. Applied egg-rr 81.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-48}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.15e-48)
    (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)
    (/
     2.0
     (pow
      (*
       (* t_m (pow (cbrt l) -2.0))
       (cbrt (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.15e-48) {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	} else {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.15e-48) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	} else {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.15e-48)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.15e-48], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.15e-48

   1. Initial program 54.7%

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

      1. associate-/r* 54.7%

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

      2. sqr-neg 54.7%

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

      3. associate-*l* 49.6%

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

      4. sqr-neg 49.6%

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

      5. associate-/r* 55.5%

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

      6. associate-+l+ 55.5%

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

      7. unpow2 55.5%

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

      8. times-frac 39.9%

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

      9. sqr-neg 39.9%

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

      10. times-frac 55.5%

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

      11. unpow2 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in k around 0 53.1%

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

      1. add-cube-cbrt 53.1%

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

      2. pow3 53.1%

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

      3. associate-/r* 50.9%

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

      4. cbrt-div 51.0%

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

      5. rem-cbrt-cube 57.5%

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

   7. Applied egg-rr 57.5%

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

      1. pow2 57.5%

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

      2. cbrt-div 61.5%

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

      3. unpow2 61.5%

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

      4. cbrt-prod 68.3%

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

      5. unpow2 68.3%

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

      6. cbrt-prod 76.7%

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

      7. pow2 76.7%

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

   9. Applied egg-rr 76.7%

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

   if 2.15e-48 < k

   1. Initial program 43.8%

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

      1. associate-*l* 43.8%

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

      2. sqr-neg 43.8%

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

      3. sqr-neg 43.8%

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

      4. associate-/r* 54.8%

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

      5. distribute-rgt-in 54.8%

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

      6. unpow2 54.8%

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

      7. times-frac 41.2%

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

      8. sqr-neg 41.2%

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

      9. times-frac 54.8%

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

      10. unpow2 54.8%

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

      11. distribute-rgt-in 54.8%

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

   3. Simplified 54.8%

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

      1. add-cube-cbrt 54.7%

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

      2. pow3 54.7%

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

      3. *-commutative 54.7%

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

      4. cbrt-prod 54.6%

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

      5. associate-/l/ 43.7%

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

      6. cbrt-div 43.6%

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

      7. rem-cbrt-cube 48.1%

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

      8. cbrt-unprod 67.2%

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

      9. pow2 67.2%

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

   6. Applied egg-rr 67.2%

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

   8. Applied egg-rr 75.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-48}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot {\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-143)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (sin k)
      (*
       (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
       (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.2e-143) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / (sin(k) * ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((t_m * pow(cbrt(l), -2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.2e-143) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.2e-143)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-143], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * 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[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.1999999999999996e-143

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 7.1999999999999996e-143 < t

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. sqr-neg 55.9%

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

      3. sqr-neg 55.9%

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

      4. associate-/r* 62.1%

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

      5. distribute-rgt-in 62.1%

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

      6. unpow2 62.1%

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

      7. times-frac 56.1%

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

      8. sqr-neg 56.1%

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

      9. times-frac 62.1%

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

      10. unpow2 62.1%

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

      11. distribute-rgt-in 62.1%

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

   3. Simplified 62.1%

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

      1. add-cube-cbrt 62.0%

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

      2. pow3 62.0%

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

      3. *-commutative 62.0%

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

      4. cbrt-prod 61.9%

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

      5. associate-/l/ 55.7%

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

      6. cbrt-div 55.7%

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

      7. rem-cbrt-cube 61.6%

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

      8. cbrt-unprod 81.3%

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

      9. pow2 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. associate-*r/ 81.2%

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

   8. Applied egg-rr 81.2%

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

      1. add-cube-cbrt 81.1%

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

      2. pow3 81.1%

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

      3. div-inv 81.2%

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

      4. pow-flip 81.1%

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

      5. metadata-eval 81.1%

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

   10. Applied egg-rr 81.1%

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

       1. expm1-log1p-u 62.3%

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

       2. expm1-udef 46.9%

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

   12. Applied egg-rr 46.0%

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

       1. expm1-def 61.4%

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

       2. expm1-log1p 80.3%

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

       3. associate-*l* 80.4%

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

   14. Simplified 80.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 9 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t_m \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right) \cdot \left(\sin k \cdot {\left(\ell \cdot {t_m}^{-1.5}\right)}^{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-143)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (if (<= t_m 1.5e+186)
      (/
       2.0
       (*
        (tan k)
        (*
         (+ 2.0 (pow (/ k t_m) 2.0))
         (* (sin k) (pow (* l (pow t_m -1.5)) -2.0)))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-143) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else if (t_m <= 1.5e+186) {
		tmp = 2.0 / (tan(k) * ((2.0 + pow((k / t_m), 2.0)) * (sin(k) * pow((l * pow(t_m, -1.5)), -2.0))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-143) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else if (t_m <= 1.5e+186) {
		tmp = 2.0 / (Math.tan(k) * ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * Math.pow((l * Math.pow(t_m, -1.5)), -2.0))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-143)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	elseif (t_m <= 1.5e+186)
		tmp = Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * (Float64(l * (t_m ^ -1.5)) ^ -2.0)))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-143], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+186], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(l * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.00000000000000001e-143

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 9.00000000000000001e-143 < t < 1.49999999999999991e186

   1. Initial program 54.1%

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

      1. associate-*l* 54.2%

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

      2. sqr-neg 54.2%

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

      3. sqr-neg 54.2%

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

      4. associate-/r* 59.2%

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

      5. distribute-rgt-in 59.2%

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

      6. unpow2 59.2%

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

      7. times-frac 58.9%

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

      8. sqr-neg 58.9%

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

      9. times-frac 59.2%

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

      10. unpow2 59.2%

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

      11. distribute-rgt-in 59.2%

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

   3. Simplified 59.2%

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

      1. add-sqr-sqrt 59.1%

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

      2. pow2 59.1%

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

      3. associate-/l/ 54.1%

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

      4. sqrt-div 54.1%

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

      5. sqrt-pow1 60.5%

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

      6. metadata-eval 60.5%

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

      7. sqrt-unprod 30.8%

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

      8. add-sqr-sqrt 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. clear-num 76.7%

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

      2. inv-pow 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. expm1-log1p-u 68.4%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. expm1-udef 48.0%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({\left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} - 1}} \]

   10. Applied egg-rr 48.0%

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

       1. expm1-def 68.4%

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

       2. expm1-log1p 76.7%

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

       3. associate-*l* 76.7%

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

   12. Simplified 76.7%

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

   if 1.49999999999999991e186 < t

   1. Initial program 60.5%

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

      1. associate-/r* 60.5%

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

      2. sqr-neg 60.5%

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

      3. associate-*l* 52.5%

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

      4. sqr-neg 52.5%

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

      5. associate-/r* 61.7%

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

      6. associate-+l+ 61.7%

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

      7. unpow2 61.7%

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

      8. times-frac 40.7%

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

      9. sqr-neg 40.7%

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

      10. times-frac 61.7%

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

      11. unpow2 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in k around 0 52.5%

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

      1. add-cube-cbrt 52.5%

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

      2. pow3 52.5%

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

      3. associate-/r* 44.0%

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

      4. cbrt-div 44.0%

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

      5. rem-cbrt-cube 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. pow2 44.8%

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

      2. cbrt-div 60.2%

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

      3. unpow2 60.2%

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

      4. cbrt-prod 73.0%

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

      5. unpow2 73.0%

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

      6. cbrt-prod 80.8%

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

      7. pow2 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {\left(\ell \cdot {t}^{-1.5}\right)}^{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t_m \leq 2.7 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-142)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (if (<= t_m 2.7e+186)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-142) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else if (t_m <= 2.7e+186) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-142) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else if (t_m <= 2.7e+186) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-142)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	elseif (t_m <= 2.7e+186)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-142], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e+186], 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[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1e-142

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 1e-142 < t < 2.6999999999999999e186

   1. Initial program 54.1%

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

      1. associate-*l* 54.2%

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

      2. sqr-neg 54.2%

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

      3. sqr-neg 54.2%

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

      4. associate-/r* 59.2%

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

      5. distribute-rgt-in 59.2%

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

      6. unpow2 59.2%

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

      7. times-frac 58.9%

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

      8. sqr-neg 58.9%

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

      9. times-frac 59.2%

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

      10. unpow2 59.2%

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

      11. distribute-rgt-in 59.2%

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

   3. Simplified 59.2%

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

      1. add-sqr-sqrt 59.1%

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

      2. pow2 59.1%

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

      3. associate-/l/ 54.1%

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

      4. sqrt-div 54.1%

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

      5. sqrt-pow1 60.5%

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

      6. metadata-eval 60.5%

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

      7. sqrt-unprod 30.8%

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

      8. add-sqr-sqrt 76.7%

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

   6. Applied egg-rr 76.7%

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

   if 2.6999999999999999e186 < t

   1. Initial program 60.5%

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

      1. associate-/r* 60.5%

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

      2. sqr-neg 60.5%

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

      3. associate-*l* 52.5%

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

      4. sqr-neg 52.5%

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

      5. associate-/r* 61.7%

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

      6. associate-+l+ 61.7%

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

      7. unpow2 61.7%

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

      8. times-frac 40.7%

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

      9. sqr-neg 40.7%

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

      10. times-frac 61.7%

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

      11. unpow2 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in k around 0 52.5%

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

      1. add-cube-cbrt 52.5%

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

      2. pow3 52.5%

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

      3. associate-/r* 44.0%

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

      4. cbrt-div 44.0%

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

      5. rem-cbrt-cube 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. pow2 44.8%

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

      2. cbrt-div 60.2%

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

      3. unpow2 60.2%

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

      4. cbrt-prod 73.0%

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

      5. unpow2 73.0%

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

      6. cbrt-prod 80.8%

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

      7. pow2 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left({\left(t_m \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-143)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (* (sin k) (* (pow (* t_m (sqrt (/ 1.0 l))) 2.0) (/ t_m l))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-143) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * (pow((t_m * sqrt((1.0 / l))), 2.0) * (t_m / l))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-143) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / (l ** 2.0d0)) * ((sin(k) ** 2.0d0) / cos(k)))
    else
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * (sin(k) * (((t_m * sqrt((1.0d0 / l))) ** 2.0d0) * (t_m / l))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-143) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * (Math.pow((t_m * Math.sqrt((1.0 / l))), 2.0) * (t_m / l))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-143:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.pow(l, 2.0)) * (math.pow(math.sin(k), 2.0) / math.cos(k)))
	else:
		tmp = 2.0 / ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * (math.sin(k) * (math.pow((t_m * math.sqrt((1.0 / l))), 2.0) * (t_m / l))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-143)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64((Float64(t_m * sqrt(Float64(1.0 / l))) ^ 2.0) * Float64(t_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-143)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / (l ^ 2.0)) * ((sin(k) ^ 2.0) / cos(k)));
	else
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * (sin(k) * (((t_m * sqrt((1.0 / l))) ^ 2.0) * (t_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-143], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.30000000000000011e-143

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 2.30000000000000011e-143 < t

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. sqr-neg 55.9%

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

      3. sqr-neg 55.9%

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

      4. associate-/r* 62.1%

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

      5. distribute-rgt-in 62.1%

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

      6. unpow2 62.1%

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

      7. times-frac 56.1%

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

      8. sqr-neg 56.1%

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

      9. times-frac 62.1%

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

      10. unpow2 62.1%

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

      11. distribute-rgt-in 62.1%

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

   3. Simplified 62.1%

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

      1. associate-/l/ 55.9%

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

      2. unpow3 55.9%

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

      3. times-frac 68.5%

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

      4. pow2 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. add-sqr-sqrt 24.3%

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

      2. pow2 24.3%

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

      3. div-inv 24.2%

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

      4. sqrt-prod 24.2%

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

      5. unpow2 24.2%

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

      6. sqrt-prod 25.3%

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

      7. add-sqr-sqrt 25.3%

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

   8. Applied egg-rr 35.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left({\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.5 \cdot 10^{-143}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t_m \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t_m \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t_m}{\ell} \cdot \frac{{t_m}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-143)
    (*
     2.0
     (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 4.5e+152)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l)))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-143) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 4.5e+152) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-143) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 4.5e+152) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.5e-143)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 4.5e+152)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l)))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.5e-143], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+152], 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[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.50000000000000041e-143

   1. Initial program 49.2%

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

      1. associate-/r* 49.2%

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

      2. sqr-neg 49.2%

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

      3. associate-*l* 46.6%

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

      4. sqr-neg 46.6%

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

      5. associate-/r* 54.6%

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

      6. associate-+l+ 54.6%

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

      7. unpow2 54.6%

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

      8. times-frac 34.7%

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

      9. sqr-neg 34.7%

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

      10. times-frac 54.6%

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

      11. unpow2 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. times-frac 62.7%

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

   7. Simplified 62.7%

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

   if 5.50000000000000041e-143 < t < 4.5000000000000001e152

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. sqr-neg 58.2%

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

      3. sqr-neg 58.2%

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

      4. associate-/r* 63.6%

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

      5. distribute-rgt-in 63.6%

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

      6. unpow2 63.6%

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

      7. times-frac 63.5%

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

      8. sqr-neg 63.5%

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

      9. times-frac 63.6%

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

      10. unpow2 63.6%

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

      11. distribute-rgt-in 63.6%

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

   3. Simplified 63.6%

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

      1. associate-/l/ 58.2%

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

      2. unpow3 58.1%

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

      3. times-frac 73.7%

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

      4. pow2 73.7%

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

   6. Applied egg-rr 73.7%

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

   if 4.5000000000000001e152 < t

   1. Initial program 52.0%

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

      1. associate-/r* 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 42.9%

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

      4. sqr-neg 42.9%

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

      5. associate-/r* 50.6%

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

      6. associate-+l+ 50.6%

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

      7. unpow2 50.6%

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

      8. times-frac 34.1%

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

      9. sqr-neg 34.1%

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

      10. times-frac 50.6%

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

      11. unpow2 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in k around 0 42.9%

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

      1. add-cube-cbrt 42.9%

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

      2. pow3 42.9%

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

      3. associate-/r* 36.4%

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

      4. cbrt-div 36.4%

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

      5. rem-cbrt-cube 37.2%

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

   7. Applied egg-rr 37.2%

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

      1. pow2 37.2%

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

      2. cbrt-div 51.7%

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

      3. unpow2 51.7%

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

      4. cbrt-prod 64.8%

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

      5. unpow2 64.8%

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

      6. cbrt-prod 73.8%

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

      7. pow2 73.8%

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

   9. Applied egg-rr 73.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-143}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t_m \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t_m}{\ell} \cdot \frac{{t_m}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-142)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (if (<= t_m 8.5e+151)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l)))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-142) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else if (t_m <= 8.5e+151) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-142) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else if (t_m <= 8.5e+151) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-142)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	elseif (t_m <= 8.5e+151)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l)))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-142], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+151], 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[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.10000000000000008e-142

   1. Initial program 49.2%

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

      1. associate-*l* 49.2%

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

      2. sqr-neg 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-/r* 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. unpow2 57.3%

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

      7. times-frac 37.4%

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

      8. sqr-neg 37.4%

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

      9. times-frac 57.3%

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

      10. unpow2 57.3%

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

      11. distribute-rgt-in 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. times-frac 65.2%

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

   7. Simplified 65.2%

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

   if 1.10000000000000008e-142 < t < 8.50000000000000051e151

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. sqr-neg 58.2%

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

      3. sqr-neg 58.2%

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

      4. associate-/r* 63.6%

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

      5. distribute-rgt-in 63.6%

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

      6. unpow2 63.6%

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

      7. times-frac 63.5%

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

      8. sqr-neg 63.5%

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

      9. times-frac 63.6%

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

      10. unpow2 63.6%

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

      11. distribute-rgt-in 63.6%

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

   3. Simplified 63.6%

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

      1. associate-/l/ 58.2%

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

      2. unpow3 58.1%

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

      3. times-frac 73.7%

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

      4. pow2 73.7%

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

   6. Applied egg-rr 73.7%

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

   if 8.50000000000000051e151 < t

   1. Initial program 52.0%

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

      1. associate-/r* 52.0%

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

      2. sqr-neg 52.0%

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

      3. associate-*l* 42.9%

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

      4. sqr-neg 42.9%

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

      5. associate-/r* 50.6%

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

      6. associate-+l+ 50.6%

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

      7. unpow2 50.6%

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

      8. times-frac 34.1%

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

      9. sqr-neg 34.1%

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

      10. times-frac 50.6%

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

      11. unpow2 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in k around 0 42.9%

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

      1. add-cube-cbrt 42.9%

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

      2. pow3 42.9%

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

      3. associate-/r* 36.4%

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

      4. cbrt-div 36.4%

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

      5. rem-cbrt-cube 37.2%

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

   7. Applied egg-rr 37.2%

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

      1. pow2 37.2%

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

      2. cbrt-div 51.7%

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

      3. unpow2 51.7%

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

      4. cbrt-prod 64.8%

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

      5. unpow2 64.8%

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

      6. cbrt-prod 73.8%

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

      7. pow2 73.8%

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

   9. Applied egg-rr 73.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{\frac{{k}^{2}}{{\ell}^{2}}}}}{t_m}\right)}^{3}\\ \mathbf{elif}\;t_m \leq 1.55 \cdot 10^{+113}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t_m \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-142)
    (pow (/ (/ 1.0 (cbrt (/ (pow k 2.0) (pow l 2.0)))) t_m) 3.0)
    (if (<= t_m 1.55e+113)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (* (sin k) (/ (/ (pow t_m 3.0) l) l))))
      (/
       2.0
       (*
        (* (sin k) (* (pow (* t_m (sqrt (/ 1.0 l))) 2.0) (/ t_m l)))
        (* 2.0 k)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-142) {
		tmp = pow(((1.0 / cbrt((pow(k, 2.0) / pow(l, 2.0)))) / t_m), 3.0);
	} else if (t_m <= 1.55e+113) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * ((pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = 2.0 / ((sin(k) * (pow((t_m * sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-142) {
		tmp = Math.pow(((1.0 / Math.cbrt((Math.pow(k, 2.0) / Math.pow(l, 2.0)))) / t_m), 3.0);
	} else if (t_m <= 1.55e+113) {
		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) / l)));
	} else {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m * Math.sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e-142)
		tmp = Float64(Float64(1.0 / cbrt(Float64((k ^ 2.0) / (l ^ 2.0)))) / t_m) ^ 3.0;
	elseif (t_m <= 1.55e+113)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m * sqrt(Float64(1.0 / l))) ^ 2.0) * Float64(t_m / l))) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-142], N[Power[N[(N[(1.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$m, 1.55e+113], 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[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.2500000000000001e-142

   1. Initial program 49.2%

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

      1. associate-/r* 49.2%

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

      2. sqr-neg 49.2%

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

      3. associate-*l* 46.6%

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

      4. sqr-neg 46.6%

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

      5. associate-/r* 54.6%

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

      6. associate-+l+ 54.6%

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

      7. unpow2 54.6%

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

      8. times-frac 34.7%

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

      9. sqr-neg 34.7%

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

      10. times-frac 54.6%

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

      11. unpow2 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in k around 0 51.7%

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

      1. add-cube-cbrt 51.6%

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

      2. pow3 51.6%

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

      3. associate-/r* 50.6%

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

      4. cbrt-div 50.6%

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

      5. rem-cbrt-cube 59.6%

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

   7. Applied egg-rr 59.6%

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

      1. pow2 59.6%

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

      2. clear-num 59.5%

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

      3. cbrt-div 59.6%

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

      4. metadata-eval 59.6%

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

      5. pow2 59.6%

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

   9. Applied egg-rr 59.6%

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

   if 1.2500000000000001e-142 < t < 1.54999999999999996e113

   1. Initial program 61.0%

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

      1. associate-*l* 61.0%

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

      2. sqr-neg 61.0%

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

      3. sqr-neg 61.0%

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

      4. associate-/r* 67.3%

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

      5. distribute-rgt-in 67.3%

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

      6. unpow2 67.3%

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

      7. times-frac 67.3%

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

      8. sqr-neg 67.3%

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

      9. times-frac 67.3%

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

      10. unpow2 67.3%

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

      11. distribute-rgt-in 67.3%

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

   3. Simplified 67.3%

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

   if 1.54999999999999996e113 < t

   1. Initial program 50.0%

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

      1. associate-*l* 50.0%

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

      2. sqr-neg 50.0%

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

      3. sqr-neg 50.0%

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

      4. associate-/r* 56.2%

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

      5. distribute-rgt-in 56.2%

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

      6. unpow2 56.2%

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

      7. times-frac 43.3%

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

      8. sqr-neg 43.3%

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

      9. times-frac 56.2%

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

      10. unpow2 56.2%

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

      11. distribute-rgt-in 56.2%

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

   3. Simplified 56.2%

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

      1. associate-/l/ 50.0%

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

      2. unpow3 50.0%

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

      3. times-frac 67.7%

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

      4. pow2 67.7%

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

   6. Applied egg-rr 67.7%

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

   7. Taylor expanded in k around 0 60.9%

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

      1. add-sqr-sqrt 29.9%

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

      2. pow2 29.9%

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

      3. div-inv 29.9%

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

      4. sqrt-prod 29.9%

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

      5. unpow2 29.9%

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

      6. sqrt-prod 32.1%

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

      7. add-sqr-sqrt 32.2%

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

   9. Applied egg-rr 32.2%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{\frac{{k}^{2}}{{\ell}^{2}}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+113}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t_m \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+48}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{t_m}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(k \cdot {t_m}^{1.5}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.8e-139)
    (/
     2.0
     (*
      (* (sin k) (* (pow (* t_m (sqrt (/ 1.0 l))) 2.0) (/ t_m l)))
      (* 2.0 k)))
    (if (<= l 4.8e+48)
      (pow (/ (cbrt (/ (pow l 2.0) (pow k 2.0))) t_m) 3.0)
      (if (<= l 6.6e+136)
        (/ (pow l 2.0) (pow (* k (pow t_m 1.5)) 2.0))
        (/
         2.0
         (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* (sin k) (tan k))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.8e-139) {
		tmp = 2.0 / ((sin(k) * (pow((t_m * sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if (l <= 4.8e+48) {
		tmp = pow((cbrt((pow(l, 2.0) / pow(k, 2.0))) / t_m), 3.0);
	} else if (l <= 6.6e+136) {
		tmp = pow(l, 2.0) / pow((k * pow(t_m, 1.5)), 2.0);
	} else {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (sin(k) * tan(k))));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.8e-139) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m * Math.sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if (l <= 4.8e+48) {
		tmp = Math.pow((Math.cbrt((Math.pow(l, 2.0) / Math.pow(k, 2.0))) / t_m), 3.0);
	} else if (l <= 6.6e+136) {
		tmp = Math.pow(l, 2.0) / Math.pow((k * Math.pow(t_m, 1.5)), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (Math.sin(k) * Math.tan(k))));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 2.8e-139)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m * sqrt(Float64(1.0 / l))) ^ 2.0) * Float64(t_m / l))) * Float64(2.0 * k)));
	elseif (l <= 4.8e+48)
		tmp = Float64(cbrt(Float64((l ^ 2.0) / (k ^ 2.0))) / t_m) ^ 3.0;
	elseif (l <= 6.6e+136)
		tmp = Float64((l ^ 2.0) / (Float64(k * (t_m ^ 1.5)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(sin(k) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 2.8e-139], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e+48], N[Power[N[(N[Power[N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, 6.6e+136], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if l < 2.7999999999999999e-139

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. sqr-neg 53.0%

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

      3. sqr-neg 53.0%

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

      4. associate-/r* 61.6%

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

      5. distribute-rgt-in 61.6%

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

      6. unpow2 61.6%

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

      7. times-frac 46.2%

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

      8. sqr-neg 46.2%

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

      9. times-frac 61.6%

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

      10. unpow2 61.6%

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

      11. distribute-rgt-in 61.6%

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

   3. Simplified 61.6%

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

      1. associate-/l/ 53.0%

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

      2. unpow3 53.0%

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

      3. times-frac 68.4%

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

      4. pow2 68.4%

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

   6. Applied egg-rr 68.4%

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

   7. Taylor expanded in k around 0 65.3%

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

      1. add-sqr-sqrt 17.5%

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

      2. pow2 17.5%

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

      3. div-inv 17.5%

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

      4. sqrt-prod 16.0%

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

      5. unpow2 16.0%

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

      6. sqrt-prod 7.7%

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

      7. add-sqr-sqrt 16.1%

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

   9. Applied egg-rr 16.1%

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

   if 2.7999999999999999e-139 < l < 4.8000000000000002e48

   1. Initial program 64.7%

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

      1. associate-/r* 64.7%

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

      2. sqr-neg 64.7%

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

      3. associate-*l* 59.0%

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

      4. sqr-neg 59.0%

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

      5. associate-/r* 59.0%

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

      6. associate-+l+ 59.0%

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

      7. unpow2 59.0%

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

      8. times-frac 49.9%

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

      9. sqr-neg 49.9%

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

      10. times-frac 59.0%

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

      11. unpow2 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in k around 0 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. associate-/r* 59.4%

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

      4. cbrt-div 59.4%

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

      5. rem-cbrt-cube 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 4.8000000000000002e48 < l < 6.59999999999999984e136

   1. Initial program 62.6%

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

      1. associate-/r* 62.6%

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

      2. sqr-neg 62.6%

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

      3. associate-*l* 62.5%

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

      4. sqr-neg 62.5%

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

      5. associate-/r* 62.4%

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

      6. associate-+l+ 62.4%

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

      7. unpow2 62.4%

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

      8. times-frac 51.3%

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

      9. sqr-neg 51.3%

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

      10. times-frac 62.4%

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

      11. unpow2 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in k around 0 57.1%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

      3. sqrt-prod 18.0%

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

      4. unpow2 18.0%

         \[\leadsto \frac{{\ell}^{2}}{{\left(\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{t}^{3}}\right)}^{2}} \]

      5. sqrt-prod 0.7%

         \[\leadsto \frac{{\ell}^{2}}{{\left(\color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \sqrt{{t}^{3}}\right)}^{2}} \]

      6. add-sqr-sqrt 18.1%

         \[\leadsto \frac{{\ell}^{2}}{{\left(\color{blue}{k} \cdot \sqrt{{t}^{3}}\right)}^{2}} \]

      7. metadata-eval 18.1%

         \[\leadsto \frac{{\ell}^{2}}{{\left(k \cdot \sqrt{{t}^{\color{blue}{\left(1.5 + 1.5\right)}}}\right)}^{2}} \]

      8. pow-prod-up 18.1%

         \[\leadsto \frac{{\ell}^{2}}{{\left(k \cdot \sqrt{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}}\right)}^{2}} \]

      9. sqrt-prod 23.5%

         \[\leadsto \frac{{\ell}^{2}}{{\left(k \cdot \color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{{t}^{1.5}}\right)}\right)}^{2}} \]

      10. add-sqr-sqrt 23.5%

          \[\leadsto \frac{{\ell}^{2}}{{\left(k \cdot \color{blue}{{t}^{1.5}}\right)}^{2}} \]

   7. Applied egg-rr 23.5%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{\left(k \cdot {t}^{1.5}\right)}^{2}}} \]

   if 6.59999999999999984e136 < l

   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* 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 45.2%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 45.2%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 25.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 25.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 45.2%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 45.2%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 45.2%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 17.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 17.6%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/l/ 14.8%

         \[\leadsto \frac{2}{\left({\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 14.8%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 15.1%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 15.1%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-unprod 25.4%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 25.5%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 25.5%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in t around inf 28.3%

      \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot \frac{\sin k}{\cos k}\right)}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 25.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)\right)\right)}} \]

      2. expm1-udef 21.3%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)\right)} - 1}} \]

      3. associate-*l* 21.3%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)\right)}\right)} - 1} \]

      4. tan-quot 21.3%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(2 \cdot \color{blue}{\tan k}\right)\right)\right)} - 1} \]

      5. *-commutative 21.3%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 2\right)}\right)\right)} - 1} \]

   9. Applied egg-rr 21.3%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot 2\right)\right)\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 25.7%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot 2\right)\right)\right)\right)}} \]

       2. expm1-log1p 28.4%

          \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot 2\right)\right)}} \]

       3. associate-*r* 28.4%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot 2\right)}} \]

       4. *-commutative 28.4%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \color{blue}{\left(2 \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   11. Simplified 28.4%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+48}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{t}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(k \cdot {t}^{1.5}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t_m \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+177} \lor \neg \left(k \leq 6.5 \cdot 10^{+236}\right):\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t_m}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-7)
    (/
     2.0
     (*
      (* (sin k) (* (pow (* t_m (sqrt (/ 1.0 l))) 2.0) (/ t_m l)))
      (* 2.0 k)))
    (if (or (<= k 1.95e+177) (not (<= k 6.5e+236)))
      (pow (/ (cbrt (* (pow l 2.0) (pow k -2.0))) t_m) 3.0)
      (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-7) {
		tmp = 2.0 / ((sin(k) * (pow((t_m * sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if ((k <= 1.95e+177) || !(k <= 6.5e+236)) {
		tmp = pow((cbrt((pow(l, 2.0) * pow(k, -2.0))) / t_m), 3.0);
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-7) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m * Math.sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if ((k <= 1.95e+177) || !(k <= 6.5e+236)) {
		tmp = Math.pow((Math.cbrt((Math.pow(l, 2.0) * Math.pow(k, -2.0))) / t_m), 3.0);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-7)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m * sqrt(Float64(1.0 / l))) ^ 2.0) * Float64(t_m / l))) * Float64(2.0 * k)));
	elseif ((k <= 1.95e+177) || !(k <= 6.5e+236))
		tmp = Float64(cbrt(Float64((l ^ 2.0) * (k ^ -2.0))) / t_m) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.2e-7], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[k, 1.95e+177], N[Not[LessEqual[k, 6.5e+236]], $MachinePrecision]], N[Power[N[(N[Power[N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.2000000000000001e-7

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 54.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 54.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 45.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 45.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 54.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 54.7%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.2%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 67.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 65.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 32.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sqrt{\frac{{t}^{2}}{\ell}} \cdot \sqrt{\frac{{t}^{2}}{\ell}}\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      2. pow2 32.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\sqrt{\frac{{t}^{2}}{\ell}}\right)}^{2}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      3. div-inv 32.2%

         \[\leadsto \frac{2}{\left(\left({\left(\sqrt{\color{blue}{{t}^{2} \cdot \frac{1}{\ell}}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      4. sqrt-prod 31.5%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{\frac{1}{\ell}}\right)}}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      5. unpow2 31.5%

         \[\leadsto \frac{2}{\left(\left({\left(\sqrt{\color{blue}{t \cdot t}} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      6. sqrt-prod 15.3%

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      7. add-sqr-sqrt 33.0%

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{t} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   9. Applied egg-rr 33.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   if 2.2000000000000001e-7 < k < 1.95e177 or 6.50000000000000006e236 < k

   1. Initial program 44.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 42.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 42.5%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 42.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 42.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 54.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 54.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 54.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 42.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 42.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 54.5%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 54.5%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 42.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 42.5%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   7. Applied egg-rr 42.5%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 42.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}} \]

      2. pow3 42.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}^{3}} \]

      3. associate-/r* 42.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{{\ell}^{2}}{k \cdot k}}{{t}^{3}}}}\right)}^{3} \]

      4. cbrt-div 42.6%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{k \cdot k}}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \]

      5. pow2 42.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{{k}^{2}}}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      6. unpow3 42.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{3} \]

      7. add-cbrt-cube 51.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\color{blue}{t}}\right)}^{3} \]

      8. pow2 51.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{k \cdot k}}}}{t}\right)}^{3} \]

      9. div-inv 51.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{{\ell}^{2} \cdot \frac{1}{k \cdot k}}}}{t}\right)}^{3} \]

      10. pow2 51.2%

          \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot \frac{1}{\color{blue}{{k}^{2}}}}}{t}\right)}^{3} \]

      11. pow-flip 51.2%

          \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}}}{t}\right)}^{3} \]

      12. metadata-eval 51.2%

          \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{\color{blue}{-2}}}}{t}\right)}^{3} \]

   9. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}} \]

   if 1.95e177 < k < 6.50000000000000006e236

   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* 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 9.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 9.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      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. add-sqr-sqrt 10.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 10.0%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/l/ 9.1%

         \[\leadsto \frac{2}{\left({\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 9.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 9.3%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 9.3%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-unprod 9.4%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 18.6%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 18.6%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 19.5%

      \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+177} \lor \neg \left(k \leq 6.5 \cdot 10^{+236}\right):\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\ell}^{2} \cdot {k}^{-2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t_m \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+177}:\\ \;\;\;\;{\left(\frac{{t_2}^{0.3333333333333333}}{t_m}\right)}^{3}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+236}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{t_2}}{t_m}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow l 2.0) (pow k -2.0))))
   (*
    t_s
    (if (<= k 2.25e-7)
      (/
       2.0
       (*
        (* (sin k) (* (pow (* t_m (sqrt (/ 1.0 l))) 2.0) (/ t_m l)))
        (* 2.0 k)))
      (if (<= k 1.5e+177)
        (pow (/ (pow t_2 0.3333333333333333) t_m) 3.0)
        (if (<= k 4.6e+236)
          (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k)))
          (pow (/ (cbrt t_2) t_m) 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) * pow(k, -2.0);
	double tmp;
	if (k <= 2.25e-7) {
		tmp = 2.0 / ((sin(k) * (pow((t_m * sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if (k <= 1.5e+177) {
		tmp = pow((pow(t_2, 0.3333333333333333) / t_m), 3.0);
	} else if (k <= 4.6e+236) {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = pow((cbrt(t_2) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(l, 2.0) * Math.pow(k, -2.0);
	double tmp;
	if (k <= 2.25e-7) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m * Math.sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if (k <= 1.5e+177) {
		tmp = Math.pow((Math.pow(t_2, 0.3333333333333333) / t_m), 3.0);
	} else if (k <= 4.6e+236) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = Math.pow((Math.cbrt(t_2) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) * (k ^ -2.0))
	tmp = 0.0
	if (k <= 2.25e-7)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m * sqrt(Float64(1.0 / l))) ^ 2.0) * Float64(t_m / l))) * Float64(2.0 * k)));
	elseif (k <= 1.5e+177)
		tmp = Float64((t_2 ^ 0.3333333333333333) / t_m) ^ 3.0;
	elseif (k <= 4.6e+236)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(cbrt(t_2) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.25e-7], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+177], N[Power[N[(N[Power[t$95$2, 0.3333333333333333], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k, 4.6e+236], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[t$95$2, 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 2.2499999999999999e-7

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 54.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 54.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 45.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 45.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 54.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 54.7%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.2%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 67.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 65.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 32.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sqrt{\frac{{t}^{2}}{\ell}} \cdot \sqrt{\frac{{t}^{2}}{\ell}}\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      2. pow2 32.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\sqrt{\frac{{t}^{2}}{\ell}}\right)}^{2}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      3. div-inv 32.2%

         \[\leadsto \frac{2}{\left(\left({\left(\sqrt{\color{blue}{{t}^{2} \cdot \frac{1}{\ell}}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      4. sqrt-prod 31.5%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{\frac{1}{\ell}}\right)}}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      5. unpow2 31.5%

         \[\leadsto \frac{2}{\left(\left({\left(\sqrt{\color{blue}{t \cdot t}} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      6. sqrt-prod 15.3%

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      7. add-sqr-sqrt 33.0%

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{t} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   9. Applied egg-rr 33.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   if 2.2499999999999999e-7 < k < 1.5e177

   1. Initial program 43.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 40.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 40.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 40.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 54.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 54.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 54.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 51.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 51.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 54.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 54.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 40.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}} \]

      2. pow3 40.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}\right)}^{3}} \]

      3. associate-/r* 40.8%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}}}\right)}^{3} \]

      4. cbrt-div 40.8%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \]

      5. rem-cbrt-cube 47.3%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\color{blue}{t}}\right)}^{3} \]

   7. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{t}\right)}^{3}} \]
   8. Step-by-step derivation

      1. pow1/3 47.3%

         \[\leadsto {\left(\frac{\color{blue}{{\left(\frac{{\ell}^{2}}{{k}^{2}}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

      2. pow2 47.3%

         \[\leadsto {\left(\frac{{\left(\frac{{\ell}^{2}}{\color{blue}{k \cdot k}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      3. div-inv 47.3%

         \[\leadsto {\left(\frac{{\color{blue}{\left({\ell}^{2} \cdot \frac{1}{k \cdot k}\right)}}^{0.3333333333333333}}{t}\right)}^{3} \]

      4. pow2 47.3%

         \[\leadsto {\left(\frac{{\left({\ell}^{2} \cdot \frac{1}{\color{blue}{{k}^{2}}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      5. pow-flip 47.3%

         \[\leadsto {\left(\frac{{\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      6. metadata-eval 47.3%

         \[\leadsto {\left(\frac{{\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

   9. Applied egg-rr 47.3%

      \[\leadsto {\left(\frac{\color{blue}{{\left({\ell}^{2} \cdot {k}^{-2}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

   if 1.5e177 < k < 4.5999999999999999e236

   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* 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 9.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 9.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      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. add-sqr-sqrt 10.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 10.0%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/l/ 9.1%

         \[\leadsto \frac{2}{\left({\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 9.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 9.3%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 9.3%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-unprod 9.4%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 18.6%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 18.6%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 19.5%

      \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 4.5999999999999999e236 < k

   1. Initial program 46.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 46.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 46.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 46.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 46.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 53.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 53.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 53.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 21.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 21.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 53.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 53.9%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 46.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 46.7%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   7. Applied egg-rr 46.7%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 46.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}} \]

      2. pow3 46.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}^{3}} \]

      3. associate-/r* 46.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{{\ell}^{2}}{k \cdot k}}{{t}^{3}}}}\right)}^{3} \]

      4. cbrt-div 46.7%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{k \cdot k}}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \]

      5. pow2 46.7%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{{k}^{2}}}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      6. unpow3 46.7%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{3} \]

      7. add-cbrt-cube 60.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\color{blue}{t}}\right)}^{3} \]

      8. pow2 60.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{k \cdot k}}}}{t}\right)}^{3} \]

      9. div-inv 60.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{{\ell}^{2} \cdot \frac{1}{k \cdot k}}}}{t}\right)}^{3} \]

      10. pow2 60.2%

          \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot \frac{1}{\color{blue}{{k}^{2}}}}}{t}\right)}^{3} \]

      11. pow-flip 60.2%

          \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}}}{t}\right)}^{3} \]

      12. metadata-eval 60.2%

          \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{\color{blue}{-2}}}}{t}\right)}^{3} \]

   9. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+177}:\\ \;\;\;\;{\left(\frac{{\left({\ell}^{2} \cdot {k}^{-2}\right)}^{0.3333333333333333}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+236}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 55000 \lor \neg \left(k \leq 3.6 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t_m}^{3}}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= k 55000.0) (not (<= k 3.6e+173)))
    (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k)))
    (/ 2.0 (/ (* 2.0 (pow k 2.0)) (* l (/ l (pow t_m 3.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((k <= 55000.0) || !(k <= 3.6e+173)) {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) / (l * (l / pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= 55000.0d0) .or. (.not. (k <= 3.6d+173))) then
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) / (l * (l / (t_m ** 3.0d0))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((k <= 55000.0) || !(k <= 3.6e+173)) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) / (l * (l / Math.pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (k <= 55000.0) or not (k <= 3.6e+173):
		tmp = 2.0 / ((math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) / (l * (l / math.pow(t_m, 3.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((k <= 55000.0) || !(k <= 3.6e+173))
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) / Float64(l * Float64(l / (t_m ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((k <= 55000.0) || ~((k <= 3.6e+173)))
		tmp = 2.0 / ((sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)) * (2.0 * k));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) / (l * (l / (t_m ^ 3.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[k, 55000.0], N[Not[LessEqual[k, 3.6e+173]], $MachinePrecision]], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 55000 or 3.6000000000000002e173 < k

   1. Initial program 53.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 53.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 53.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 53.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 59.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 59.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 59.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 59.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 59.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 59.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 27.6%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/l/ 25.1%

         \[\leadsto \frac{2}{\left({\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 25.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 28.8%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 28.8%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-unprod 16.6%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 35.4%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 35.4%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 32.8%

      \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 55000 < k < 3.6000000000000002e173

   1. Initial program 41.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 41.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 41.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 41.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 56.7%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 53.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 53.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 56.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 56.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. clear-num 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-*l/ 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

      4. *-un-lft-identity 56.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      5. associate-*r* 56.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      6. div-inv 56.6%

         \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{\ell \cdot \frac{1}{\frac{{t}^{3}}{\ell}}}}} \]

      7. clear-num 56.7%

         \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \color{blue}{\frac{\ell}{{t}^{3}}}}} \]

   6. Applied egg-rr 56.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

   7. Taylor expanded in k around 0 44.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot {k}^{2}}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 55000 \lor \neg \left(k \leq 3.6 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t}^{3}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 54000:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t_m \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t_m}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 54000.0)
    (/
     2.0
     (*
      (* (sin k) (* (pow (* t_m (sqrt (/ 1.0 l))) 2.0) (/ t_m l)))
      (* 2.0 k)))
    (if (<= k 4.4e+173)
      (/ 2.0 (/ (* 2.0 (pow k 2.0)) (* l (/ l (pow t_m 3.0)))))
      (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 54000.0) {
		tmp = 2.0 / ((sin(k) * (pow((t_m * sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if (k <= 4.4e+173) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) / (l * (l / pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 54000.0d0) then
        tmp = 2.0d0 / ((sin(k) * (((t_m * sqrt((1.0d0 / l))) ** 2.0d0) * (t_m / l))) * (2.0d0 * k))
    else if (k <= 4.4d+173) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) / (l * (l / (t_m ** 3.0d0))))
    else
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)) * (2.0d0 * k))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 54000.0) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m * Math.sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k));
	} else if (k <= 4.4e+173) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) / (l * (l / Math.pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 54000.0:
		tmp = 2.0 / ((math.sin(k) * (math.pow((t_m * math.sqrt((1.0 / l))), 2.0) * (t_m / l))) * (2.0 * k))
	elif k <= 4.4e+173:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) / (l * (l / math.pow(t_m, 3.0))))
	else:
		tmp = 2.0 / ((math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 54000.0)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m * sqrt(Float64(1.0 / l))) ^ 2.0) * Float64(t_m / l))) * Float64(2.0 * k)));
	elseif (k <= 4.4e+173)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) / Float64(l * Float64(l / (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 54000.0)
		tmp = 2.0 / ((sin(k) * (((t_m * sqrt((1.0 / l))) ^ 2.0) * (t_m / l))) * (2.0 * k));
	elseif (k <= 4.4e+173)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) / (l * (l / (t_m ^ 3.0))));
	else
		tmp = 2.0 / ((sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)) * (2.0 * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 54000.0], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.4e+173], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 54000

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 54.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 54.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 46.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 46.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 60.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 54.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 54.9%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.2%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 67.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 65.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 31.7%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sqrt{\frac{{t}^{2}}{\ell}} \cdot \sqrt{\frac{{t}^{2}}{\ell}}\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      2. pow2 31.7%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\sqrt{\frac{{t}^{2}}{\ell}}\right)}^{2}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      3. div-inv 31.7%

         \[\leadsto \frac{2}{\left(\left({\left(\sqrt{\color{blue}{{t}^{2} \cdot \frac{1}{\ell}}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      4. sqrt-prod 31.1%

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{\frac{1}{\ell}}\right)}}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      5. unpow2 31.1%

         \[\leadsto \frac{2}{\left(\left({\left(\sqrt{\color{blue}{t \cdot t}} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      6. sqrt-prod 15.0%

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

      7. add-sqr-sqrt 32.5%

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{t} \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   9. Applied egg-rr 32.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(2 \cdot k\right)} \]

   if 54000 < k < 4.4e173

   1. Initial program 41.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 41.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 41.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 41.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 56.7%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 53.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 53.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 56.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 56.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. clear-num 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-*l/ 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

      4. *-un-lft-identity 56.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      5. associate-*r* 56.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      6. div-inv 56.6%

         \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{\ell \cdot \frac{1}{\frac{{t}^{3}}{\ell}}}}} \]

      7. clear-num 56.7%

         \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \color{blue}{\frac{\ell}{{t}^{3}}}}} \]

   6. Applied egg-rr 56.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

   7. Taylor expanded in k around 0 44.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot {k}^{2}}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

   if 4.4e173 < k

   1. Initial program 38.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 38.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 38.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 38.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 50.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 50.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 50.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 15.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 15.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 50.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 50.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 50.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 12.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. pow2 12.2%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. associate-/l/ 11.6%

         \[\leadsto \frac{2}{\left({\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. sqrt-div 11.5%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. sqrt-pow1 11.6%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. metadata-eval 11.6%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. sqrt-unprod 11.7%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. add-sqr-sqrt 15.9%

         \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 15.9%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 16.2%

      \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 54000:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(t \cdot \sqrt{\frac{1}{\ell}}\right)}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 54000 \lor \neg \left(k \leq 1.55 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t_m}{\ell} \cdot \frac{{t_m}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t_m}^{3}}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= k 54000.0) (not (<= k 1.55e+174)))
    (/ 2.0 (* (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l))) (* 2.0 k)))
    (/ 2.0 (/ (* 2.0 (pow k 2.0)) (* l (/ l (pow t_m 3.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((k <= 54000.0) || !(k <= 1.55e+174)) {
		tmp = 2.0 / ((sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) / (l * (l / pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= 54000.0d0) .or. (.not. (k <= 1.55d+174))) then
        tmp = 2.0d0 / ((sin(k) * ((t_m / l) * ((t_m ** 2.0d0) / l))) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) / (l * (l / (t_m ** 3.0d0))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((k <= 54000.0) || !(k <= 1.55e+174)) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) / (l * (l / Math.pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (k <= 54000.0) or not (k <= 1.55e+174):
		tmp = 2.0 / ((math.sin(k) * ((t_m / l) * (math.pow(t_m, 2.0) / l))) * (2.0 * k))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) / (l * (l / math.pow(t_m, 3.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((k <= 54000.0) || !(k <= 1.55e+174))
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l))) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) / Float64(l * Float64(l / (t_m ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((k <= 54000.0) || ~((k <= 1.55e+174)))
		tmp = 2.0 / ((sin(k) * ((t_m / l) * ((t_m ^ 2.0) / l))) * (2.0 * k));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) / (l * (l / (t_m ^ 3.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[k, 54000.0], N[Not[LessEqual[k, 1.55e+174]], $MachinePrecision]], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 54000 or 1.55e174 < k

   1. Initial program 53.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 53.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 53.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 53.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 59.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 59.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 59.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 59.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 59.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 59.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 53.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 53.0%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 65.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 65.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 65.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 64.6%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 54000 < k < 1.55e174

   1. Initial program 41.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 41.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 41.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 41.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 56.7%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 53.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 53.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 56.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 56.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 56.7%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. clear-num 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-*l/ 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

      4. *-un-lft-identity 56.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      5. associate-*r* 56.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

      6. div-inv 56.6%

         \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{\ell \cdot \frac{1}{\frac{{t}^{3}}{\ell}}}}} \]

      7. clear-num 56.7%

         \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \color{blue}{\frac{\ell}{{t}^{3}}}}} \]

   6. Applied egg-rr 56.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

   7. Taylor expanded in k around 0 44.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot {k}^{2}}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 54000 \lor \neg \left(k \leq 1.55 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t}^{3}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t_m}^{3}}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 k) (/ k (/ (pow l 2.0) (pow t_m 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * k) * (k / (pow(l, 2.0) / pow(t_m, 3.0)))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * k) * (k / ((l ** 2.0d0) / (t_m ** 3.0d0)))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * k) * (k / (Math.pow(l, 2.0) / Math.pow(t_m, 3.0)))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * k) * (k / (math.pow(l, 2.0) / math.pow(t_m, 3.0)))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k / Float64((l ^ 2.0) / (t_m ^ 3.0))))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * k) * (k / ((l ^ 2.0) / (t_m ^ 3.0)))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k / N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 51.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. sqr-neg 51.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. sqr-neg 51.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 59.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 59.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 59.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 43.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 43.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 59.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 59.0%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 59.0%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 59.0%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ 51.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   2. unpow3 51.5%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   3. times-frac 65.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   4. pow2 65.4%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

6. Applied egg-rr 65.4%

   \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

7. Taylor expanded in k around 0 61.0%

   \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

8. Taylor expanded in k around 0 54.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
9. Step-by-step derivation

   1. associate-/l* 54.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(2 \cdot k\right)} \]

10. Simplified 54.3%

    \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(2 \cdot k\right)} \]

11. Final simplification 54.3%

    \[\leadsto \frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \]
12. Add Preprocessing

Alternative 21: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t_m}^{3}}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (* 2.0 (pow k 2.0)) (* l (/ l (pow t_m 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * pow(k, 2.0)) / (l * (l / pow(t_m, 3.0)))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k ** 2.0d0)) / (l * (l / (t_m ** 3.0d0)))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * Math.pow(k, 2.0)) / (l * (l / Math.pow(t_m, 3.0)))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * math.pow(k, 2.0)) / (l * (l / math.pow(t_m, 3.0)))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) / Float64(l * Float64(l / (t_m ^ 3.0))))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k ^ 2.0)) / (l * (l / (t_m ^ 3.0)))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 51.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. sqr-neg 51.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. sqr-neg 51.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 59.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 59.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 59.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 43.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 43.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 59.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 59.0%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 59.0%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 59.0%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 55.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   2. clear-num 55.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   3. associate-*l/ 55.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}}} \]

   4. *-un-lft-identity 55.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

   5. associate-*r* 55.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \]

   6. div-inv 55.6%

      \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{\ell \cdot \frac{1}{\frac{{t}^{3}}{\ell}}}}} \]

   7. clear-num 55.6%

      \[\leadsto \frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \color{blue}{\frac{\ell}{{t}^{3}}}}} \]

6. Applied egg-rr 55.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \frac{\ell}{{t}^{3}}}}} \]

7. Taylor expanded in k around 0 56.4%

   \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot {k}^{2}}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]

8. Final simplification 56.4%

   \[\leadsto \frac{2}{\frac{2 \cdot {k}^{2}}{\ell \cdot \frac{\ell}{{t}^{3}}}} \]
9. Add Preprocessing

Alternative 22: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{{\ell}^{2}}{{t_m}^{3} \cdot \left(k \cdot k\right)} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (pow l 2.0) (* (pow t_m 3.0) (* k k)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (pow(l, 2.0) / (pow(t_m, 3.0) * (k * k)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l ** 2.0d0) / ((t_m ** 3.0d0) * (k * k)))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (Math.pow(l, 2.0) / (Math.pow(t_m, 3.0) * (k * k)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (math.pow(l, 2.0) / (math.pow(t_m, 3.0) * (k * k)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64((l ^ 2.0) / Float64((t_m ^ 3.0) * Float64(k * k))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l ^ 2.0) / ((t_m ^ 3.0) * (k * k)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 51.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. sqr-neg 51.2%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. associate-*l* 47.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. sqr-neg 47.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. associate-/r* 54.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. associate-+l+ 54.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

   7. unpow2 54.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

   8. times-frac 39.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

   9. sqr-neg 39.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

   10. times-frac 54.9%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

   11. unpow2 54.9%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

3. Simplified 54.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 50.5%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. unpow2 50.5%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

7. Applied egg-rr 50.5%

   \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

8. Final simplification 50.5%

   \[\leadsto \frac{{\ell}^{2}}{{t}^{3} \cdot \left(k \cdot k\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))))