Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.2% → 78.9%

Time: 35.4s

Alternatives: 27

Speedup: 2.0×

• 35.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.2% 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: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-166}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot t\_2\right)\right)}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.02 \cdot 10^{+179}:\\ \;\;\;\;\frac{2}{\frac{k}{t\_m}} \cdot \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}}{\frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {k}^{-2}\right) \cdot \frac{t\_2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) -2.0)))
   (*
    t_s
    (if (<= t_m 2.25e-166)
      (/ (* 2.0 (* (pow l 2.0) (* (cos k) t_2))) (* t_m (pow k 2.0)))
      (if (<= t_m 1.02e+179)
        (*
         (/ 2.0 (/ k t_m))
         (/
          (pow
           (/ (pow (cbrt l) 2.0) (* t_m (* (cbrt (tan k)) (cbrt (sin k)))))
           3.0)
          (/ k t_m)))
        (* (* 2.0 (pow k -2.0)) (/ (* t_2 (* (pow l 2.0) (cos k))) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), -2.0);
	double tmp;
	if (t_m <= 2.25e-166) {
		tmp = (2.0 * (pow(l, 2.0) * (cos(k) * t_2))) / (t_m * pow(k, 2.0));
	} else if (t_m <= 1.02e+179) {
		tmp = (2.0 / (k / t_m)) * (pow((pow(cbrt(l), 2.0) / (t_m * (cbrt(tan(k)) * cbrt(sin(k))))), 3.0) / (k / t_m));
	} else {
		tmp = (2.0 * pow(k, -2.0)) * ((t_2 * (pow(l, 2.0) * cos(k))) / t_m);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), -2.0);
	double tmp;
	if (t_m <= 2.25e-166) {
		tmp = (2.0 * (Math.pow(l, 2.0) * (Math.cos(k) * t_2))) / (t_m * Math.pow(k, 2.0));
	} else if (t_m <= 1.02e+179) {
		tmp = (2.0 / (k / t_m)) * (Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * (Math.cbrt(Math.tan(k)) * Math.cbrt(Math.sin(k))))), 3.0) / (k / t_m));
	} else {
		tmp = (2.0 * Math.pow(k, -2.0)) * ((t_2 * (Math.pow(l, 2.0) * Math.cos(k))) / t_m);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ -2.0
	tmp = 0.0
	if (t_m <= 2.25e-166)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) * t_2))) / Float64(t_m * (k ^ 2.0)));
	elseif (t_m <= 1.02e+179)
		tmp = Float64(Float64(2.0 / Float64(k / t_m)) * Float64((Float64((cbrt(l) ^ 2.0) / Float64(t_m * Float64(cbrt(tan(k)) * cbrt(sin(k))))) ^ 3.0) / Float64(k / t_m)));
	else
		tmp = Float64(Float64(2.0 * (k ^ -2.0)) * Float64(Float64(t_2 * Float64((l ^ 2.0) * cos(k))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.25e-166], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.02e+179], N[(N[(2.0 / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2499999999999999e-166

   1. Initial program 35.1%

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

      1. associate-*l* 35.1%

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

      2. associate-/r* 35.1%

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

      3. sub-neg 35.1%

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

      4. distribute-rgt-in 29.6%

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

      5. unpow2 29.6%

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

      6. times-frac 19.8%

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

      7. sqr-neg 19.8%

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

      8. times-frac 29.6%

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

      9. unpow2 29.6%

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

      10. distribute-rgt-in 35.1%

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

      11. +-commutative 35.1%

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

      12. associate-+l+ 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-*r* 79.1%

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

      3. times-frac 79.1%

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

   7. Simplified 79.1%

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

      1. associate-*l/ 79.1%

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

      2. div-inv 79.1%

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

      3. pow-flip 79.1%

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

      4. metadata-eval 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. times-frac 78.6%

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

   11. Simplified 78.6%

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

       1. frac-times 79.1%

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

       2. associate-*l* 79.1%

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

   13. Applied egg-rr 79.1%

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

   if 2.2499999999999999e-166 < t < 1.0199999999999999e179

   1. Initial program 44.1%

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

      1. associate-*l* 44.1%

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

      2. associate-/r* 44.8%

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

      3. sub-neg 44.8%

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

      4. distribute-rgt-in 44.8%

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

      5. unpow2 44.8%

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

      6. times-frac 43.4%

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

      7. sqr-neg 43.4%

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

      8. times-frac 44.8%

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

      9. unpow2 44.8%

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

      10. distribute-rgt-in 44.8%

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

      11. +-commutative 44.8%

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

      12. associate-+l+ 53.4%

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

   3. Simplified 53.4%

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

      1. +-rgt-identity 53.4%

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

      2. associate-/r* 53.4%

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

      3. unpow2 53.4%

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

      4. associate-/r* 53.4%

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

      5. associate-/l/ 53.4%

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

      6. *-commutative 53.4%

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

      7. associate-/r* 53.4%

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

      8. associate-/r/ 53.4%

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

   6. Applied egg-rr 59.2%

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

      1. add-cube-cbrt 59.1%

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

      2. pow2 59.1%

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

      3. cbrt-div 59.1%

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

      4. unpow2 59.1%

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

      5. cbrt-prod 59.0%

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

      6. pow2 59.0%

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

      7. cbrt-prod 59.0%

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

      8. unpow3 59.0%

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

      9. add-cbrt-cube 59.0%

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

      10. cbrt-div 59.0%

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

      11. unpow2 59.0%

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

      12. cbrt-prod 68.7%

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

      13. pow2 68.7%

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

   8. Applied egg-rr 85.6%

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

      1. pow-plus 85.6%

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

      2. metadata-eval 85.6%

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

   10. Simplified 85.6%

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

       1. *-commutative 85.6%

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

       2. cbrt-prod 86.7%

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

   12. Applied egg-rr 86.7%

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

   if 1.0199999999999999e179 < t

   1. Initial program 9.1%

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

      1. associate-*l* 9.1%

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

      2. associate-/r* 9.1%

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

      3. sub-neg 9.1%

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

      4. distribute-rgt-in 9.1%

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

      5. unpow2 9.1%

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

      6. times-frac 0.0%

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

      7. sqr-neg 0.0%

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

      8. times-frac 9.1%

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

      9. unpow2 9.1%

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

      10. distribute-rgt-in 9.1%

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

      11. +-commutative 9.1%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 77.9%

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

      3. times-frac 77.9%

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

   7. Simplified 77.9%

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

      1. associate-*l/ 77.9%

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

      2. div-inv 77.8%

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

      3. pow-flip 77.9%

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

      4. metadata-eval 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. times-frac 82.2%

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

   11. Simplified 82.2%

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

       1. expm1-log1p-u 81.1%

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

       2. expm1-udef 68.4%

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

       3. div-inv 68.4%

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

       4. pow-flip 68.4%

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

       5. metadata-eval 68.4%

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

   13. Applied egg-rr 68.4%

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

       1. expm1-def 81.1%

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

       2. expm1-log1p 82.3%

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

   15. Simplified 82.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-166}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+179}:\\ \;\;\;\;\frac{2}{\frac{k}{t}} \cdot \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {k}^{-2}\right) \cdot \frac{{\sin k}^{-2} \cdot \left({\ell}^{2} \cdot \cos k\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot t\_2\right)\right)}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{+178}:\\ \;\;\;\;\frac{2}{\frac{k}{t\_m}} \cdot \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}}{\frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {k}^{-2}\right) \cdot \frac{t\_2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) -2.0)))
   (*
    t_s
    (if (<= t_m 7.8e-168)
      (/ (* 2.0 (* (pow l 2.0) (* (cos k) t_2))) (* t_m (pow k 2.0)))
      (if (<= t_m 6.4e+178)
        (*
         (/ 2.0 (/ k t_m))
         (/
          (pow (/ (pow (cbrt l) 2.0) (* t_m (cbrt (* (sin k) (tan k))))) 3.0)
          (/ k t_m)))
        (* (* 2.0 (pow k -2.0)) (/ (* t_2 (* (pow l 2.0) (cos k))) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), -2.0);
	double tmp;
	if (t_m <= 7.8e-168) {
		tmp = (2.0 * (pow(l, 2.0) * (cos(k) * t_2))) / (t_m * pow(k, 2.0));
	} else if (t_m <= 6.4e+178) {
		tmp = (2.0 / (k / t_m)) * (pow((pow(cbrt(l), 2.0) / (t_m * cbrt((sin(k) * tan(k))))), 3.0) / (k / t_m));
	} else {
		tmp = (2.0 * pow(k, -2.0)) * ((t_2 * (pow(l, 2.0) * cos(k))) / t_m);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), -2.0);
	double tmp;
	if (t_m <= 7.8e-168) {
		tmp = (2.0 * (Math.pow(l, 2.0) * (Math.cos(k) * t_2))) / (t_m * Math.pow(k, 2.0));
	} else if (t_m <= 6.4e+178) {
		tmp = (2.0 / (k / t_m)) * (Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.cbrt((Math.sin(k) * Math.tan(k))))), 3.0) / (k / t_m));
	} else {
		tmp = (2.0 * Math.pow(k, -2.0)) * ((t_2 * (Math.pow(l, 2.0) * Math.cos(k))) / t_m);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ -2.0
	tmp = 0.0
	if (t_m <= 7.8e-168)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) * t_2))) / Float64(t_m * (k ^ 2.0)));
	elseif (t_m <= 6.4e+178)
		tmp = Float64(Float64(2.0 / Float64(k / t_m)) * Float64((Float64((cbrt(l) ^ 2.0) / Float64(t_m * cbrt(Float64(sin(k) * tan(k))))) ^ 3.0) / Float64(k / t_m)));
	else
		tmp = Float64(Float64(2.0 * (k ^ -2.0)) * Float64(Float64(t_2 * Float64((l ^ 2.0) * cos(k))) / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.8e-168], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.4e+178], N[(N[(2.0 / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.80000000000000025e-168

   1. Initial program 35.1%

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

      1. associate-*l* 35.1%

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

      2. associate-/r* 35.1%

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

      3. sub-neg 35.1%

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

      4. distribute-rgt-in 29.6%

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

      5. unpow2 29.6%

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

      6. times-frac 19.8%

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

      7. sqr-neg 19.8%

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

      8. times-frac 29.6%

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

      9. unpow2 29.6%

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

      10. distribute-rgt-in 35.1%

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

      11. +-commutative 35.1%

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

      12. associate-+l+ 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-*r* 79.1%

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

      3. times-frac 79.1%

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

   7. Simplified 79.1%

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

      1. associate-*l/ 79.1%

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

      2. div-inv 79.1%

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

      3. pow-flip 79.1%

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

      4. metadata-eval 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. times-frac 78.6%

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

   11. Simplified 78.6%

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

       1. frac-times 79.1%

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

       2. associate-*l* 79.1%

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

   13. Applied egg-rr 79.1%

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

   if 7.80000000000000025e-168 < t < 6.4e178

   1. Initial program 44.1%

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

      1. associate-*l* 44.1%

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

      2. associate-/r* 44.8%

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

      3. sub-neg 44.8%

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

      4. distribute-rgt-in 44.8%

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

      5. unpow2 44.8%

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

      6. times-frac 43.4%

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

      7. sqr-neg 43.4%

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

      8. times-frac 44.8%

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

      9. unpow2 44.8%

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

      10. distribute-rgt-in 44.8%

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

      11. +-commutative 44.8%

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

      12. associate-+l+ 53.4%

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

   3. Simplified 53.4%

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

      1. +-rgt-identity 53.4%

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

      2. associate-/r* 53.4%

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

      3. unpow2 53.4%

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

      4. associate-/r* 53.4%

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

      5. associate-/l/ 53.4%

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

      6. *-commutative 53.4%

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

      7. associate-/r* 53.4%

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

      8. associate-/r/ 53.4%

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

   6. Applied egg-rr 59.2%

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

      1. add-cube-cbrt 59.1%

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

      2. pow2 59.1%

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

      3. cbrt-div 59.1%

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

      4. unpow2 59.1%

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

      5. cbrt-prod 59.0%

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

      6. pow2 59.0%

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

      7. cbrt-prod 59.0%

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

      8. unpow3 59.0%

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

      9. add-cbrt-cube 59.0%

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

      10. cbrt-div 59.0%

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

      11. unpow2 59.0%

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

      12. cbrt-prod 68.7%

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

      13. pow2 68.7%

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

   8. Applied egg-rr 85.6%

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

      1. pow-plus 85.6%

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

      2. metadata-eval 85.6%

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

   10. Simplified 85.6%

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

   if 6.4e178 < t

   1. Initial program 9.1%

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

      1. associate-*l* 9.1%

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

      2. associate-/r* 9.1%

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

      3. sub-neg 9.1%

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

      4. distribute-rgt-in 9.1%

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

      5. unpow2 9.1%

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

      6. times-frac 0.0%

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

      7. sqr-neg 0.0%

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

      8. times-frac 9.1%

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

      9. unpow2 9.1%

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

      10. distribute-rgt-in 9.1%

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

      11. +-commutative 9.1%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 77.9%

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

      3. times-frac 77.9%

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

   7. Simplified 77.9%

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

      1. associate-*l/ 77.9%

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

      2. div-inv 77.8%

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

      3. pow-flip 77.9%

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

      4. metadata-eval 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. times-frac 82.2%

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

   11. Simplified 82.2%

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

       1. expm1-log1p-u 81.1%

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

       2. expm1-udef 68.4%

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

       3. div-inv 68.4%

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

       4. pow-flip 68.4%

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

       5. metadata-eval 68.4%

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

   13. Applied egg-rr 68.4%

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

       1. expm1-def 81.1%

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

       2. expm1-log1p 82.3%

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

   15. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+178}:\\ \;\;\;\;\frac{2}{\frac{k}{t}} \cdot \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {k}^{-2}\right) \cdot \frac{{\sin k}^{-2} \cdot \left({\ell}^{2} \cdot \cos k\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{\tan k \cdot \frac{\sin k}{\ell}} \cdot \frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}}{{\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t\_m \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{t\_m}} \cdot \frac{\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{\sin k \cdot \tan k}}{\frac{k}{t\_m}}\\ \end{array} \end{array} \]

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 l) 0.0)
    (/
     (/ 2.0 (pow (* (cbrt (* (tan k) (/ (sin k) l))) (/ t_m (cbrt l))) 3.0))
     (pow (/ k t_m) 2.0))
    (if (<= (* l l) 1e+225)
      (/
       (* 2.0 (* (pow l 2.0) (* (cos k) (pow (sin k) -2.0))))
       (* t_m (pow k 2.0)))
      (*
       (/ 2.0 (/ k t_m))
       (/
        (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (* (sin k) (tan k)))
        (/ k t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / pow((cbrt((tan(k) * (sin(k) / l))) * (t_m / cbrt(l))), 3.0)) / pow((k / t_m), 2.0);
	} else if ((l * l) <= 1e+225) {
		tmp = (2.0 * (pow(l, 2.0) * (cos(k) * pow(sin(k), -2.0)))) / (t_m * pow(k, 2.0));
	} else {
		tmp = (2.0 / (k / t_m)) * ((pow((pow(cbrt(l), 2.0) / t_m), 3.0) / (sin(k) * tan(k))) / (k / t_m));
	}
	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 * l) <= 0.0) {
		tmp = (2.0 / Math.pow((Math.cbrt((Math.tan(k) * (Math.sin(k) / l))) * (t_m / Math.cbrt(l))), 3.0)) / Math.pow((k / t_m), 2.0);
	} else if ((l * l) <= 1e+225) {
		tmp = (2.0 * (Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.sin(k), -2.0)))) / (t_m * Math.pow(k, 2.0));
	} else {
		tmp = (2.0 / (k / t_m)) * ((Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / (Math.sin(k) * Math.tan(k))) / (k / t_m));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(sin(k) / l))) * Float64(t_m / cbrt(l))) ^ 3.0)) / (Float64(k / t_m) ^ 2.0));
	elseif (Float64(l * l) <= 1e+225)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) * (sin(k) ^ -2.0)))) / Float64(t_m * (k ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(k / t_m)) * Float64(Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / Float64(sin(k) * tan(k))) / Float64(k / t_m)));
	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[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+225], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-*l/ 32.4%

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

      4. associate-/l* 32.4%

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

      5. +-commutative 32.4%

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

      6. unpow2 32.4%

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

      7. sqr-neg 32.4%

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

      8. distribute-frac-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. unpow2 32.4%

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

      11. associate--l+ 48.1%

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

      12. metadata-eval 48.1%

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

      13. +-rgt-identity 48.1%

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

      14. unpow2 48.1%

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

      15. distribute-frac-neg 48.1%

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

      16. distribute-frac-neg 48.1%

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

      17. sqr-neg 48.1%

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

      18. unpow2 48.1%

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

   3. Simplified 48.1%

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

      1. associate-/l* 64.3%

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

      2. associate-/r/ 66.4%

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

      3. associate-/l* 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. add-cube-cbrt 66.2%

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

      2. pow3 66.2%

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

      3. *-commutative 66.2%

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

      4. cbrt-prod 66.1%

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

      5. associate-/r/ 66.2%

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

      6. cbrt-div 66.2%

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

      7. unpow3 66.1%

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

      8. add-cbrt-cube 80.0%

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

   8. Applied egg-rr 80.0%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999928e224

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-/r* 38.8%

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

      3. sub-neg 38.8%

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

      4. distribute-rgt-in 35.2%

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

      5. unpow2 35.2%

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

      6. times-frac 24.5%

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

      7. sqr-neg 24.5%

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

      8. times-frac 35.2%

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

      9. unpow2 35.2%

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

      10. distribute-rgt-in 38.8%

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

      11. +-commutative 38.8%

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

      12. associate-+l+ 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in t around 0 93.8%

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

      1. associate-*r/ 93.9%

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

      2. associate-*r* 93.9%

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

      3. times-frac 93.9%

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

   7. Simplified 93.9%

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

      1. associate-*l/ 93.9%

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

      2. div-inv 93.8%

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

      3. pow-flip 93.9%

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

      4. metadata-eval 93.9%

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

   9. Applied egg-rr 93.9%

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

       1. times-frac 91.3%

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

   11. Simplified 91.3%

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

       1. frac-times 93.9%

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

       2. associate-*l* 93.9%

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

   13. Applied egg-rr 93.9%

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

   if 9.99999999999999928e224 < (*.f64 l l)

   1. Initial program 33.4%

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

      1. associate-*l* 33.4%

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

      2. associate-/r* 33.4%

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

      3. sub-neg 33.4%

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

      4. distribute-rgt-in 27.1%

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

      5. unpow2 27.1%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 27.1%

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

      9. unpow2 27.1%

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

      10. distribute-rgt-in 33.4%

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

      11. +-commutative 33.4%

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

      12. associate-+l+ 33.6%

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

   3. Simplified 33.6%

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

      1. +-rgt-identity 33.6%

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

      2. associate-/r* 33.6%

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

      3. unpow2 33.6%

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

      4. associate-/r* 33.6%

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

      5. associate-/l/ 33.6%

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

      6. *-commutative 33.6%

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

      7. associate-/r* 33.6%

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

      8. associate-/r/ 33.6%

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

   6. Applied egg-rr 50.1%

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

      1. add-cube-cbrt 50.0%

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

      2. pow2 50.0%

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

      3. cbrt-div 50.0%

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

      4. unpow2 50.0%

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

      5. cbrt-prod 50.0%

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

      6. pow2 50.0%

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

      7. cbrt-prod 50.0%

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

      8. unpow3 50.0%

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

      9. add-cbrt-cube 50.0%

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

      10. cbrt-div 50.0%

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

      11. unpow2 50.0%

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

      12. cbrt-prod 55.2%

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

      13. pow2 55.2%

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

   8. Applied egg-rr 74.8%

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

      1. pow-plus 74.8%

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

      2. metadata-eval 74.8%

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

      3. associate-/r* 74.8%

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

      4. cube-div 72.3%

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

      5. rem-cube-cbrt 72.3%

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

   10. Simplified 72.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{\tan k \cdot \frac{\sin k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{t}} \cdot \frac{\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{\sin k \cdot \tan k}}{\frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 10^{+225}\right):\\ \;\;\;\;\frac{2}{\frac{k}{t\_m}} \cdot \frac{\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{\sin k \cdot \tan k}}{\frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t\_m \cdot {k}^{2}}\\ \end{array} \end{array} \]

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 (<= (* l l) 0.0) (not (<= (* l l) 1e+225)))
    (*
     (/ 2.0 (/ k t_m))
     (/
      (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (* (sin k) (tan k)))
      (/ k t_m)))
    (/
     (* 2.0 (* (pow l 2.0) (* (cos k) (pow (sin k) -2.0))))
     (* t_m (pow k 2.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 1e+225)) {
		tmp = (2.0 / (k / t_m)) * ((pow((pow(cbrt(l), 2.0) / t_m), 3.0) / (sin(k) * tan(k))) / (k / t_m));
	} else {
		tmp = (2.0 * (pow(l, 2.0) * (cos(k) * pow(sin(k), -2.0)))) / (t_m * pow(k, 2.0));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 1e+225)) {
		tmp = (2.0 / (k / t_m)) * ((Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / (Math.sin(k) * Math.tan(k))) / (k / t_m));
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.sin(k), -2.0)))) / (t_m * Math.pow(k, 2.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((Float64(l * l) <= 0.0) || !(Float64(l * l) <= 1e+225))
		tmp = Float64(Float64(2.0 / Float64(k / t_m)) * Float64(Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / Float64(sin(k) * tan(k))) / Float64(k / t_m)));
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) * (sin(k) ^ -2.0)))) / Float64(t_m * (k ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(l * l), $MachinePrecision], 1e+225]], $MachinePrecision]], N[(N[(2.0 / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0 or 9.99999999999999928e224 < (*.f64 l l)

   1. Initial program 33.0%

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

      1. associate-*l* 33.0%

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

      2. associate-/r* 33.0%

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

      3. sub-neg 33.0%

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

      4. distribute-rgt-in 29.5%

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

      5. unpow2 29.5%

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

      6. times-frac 24.6%

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

      7. sqr-neg 24.6%

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

      8. times-frac 29.5%

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

      9. unpow2 29.5%

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

      10. distribute-rgt-in 33.0%

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

      11. +-commutative 33.0%

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

      12. associate-+l+ 40.1%

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

   3. Simplified 40.1%

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

      1. +-rgt-identity 40.1%

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

      2. associate-/r* 40.1%

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

      3. unpow2 40.1%

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

      4. associate-/r* 40.1%

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

      5. associate-/l/ 40.1%

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

      6. *-commutative 40.1%

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

      7. associate-/r* 40.1%

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

      8. associate-/r/ 40.1%

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

   6. Applied egg-rr 51.3%

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

      1. add-cube-cbrt 51.2%

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

      2. pow2 51.2%

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

      3. cbrt-div 51.2%

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

      4. unpow2 51.2%

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

      5. cbrt-prod 51.2%

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

      6. pow2 51.2%

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

      7. cbrt-prod 51.2%

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

      8. unpow3 51.2%

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

      9. add-cbrt-cube 51.2%

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

      10. cbrt-div 51.2%

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

      11. unpow2 51.2%

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

      12. cbrt-prod 62.3%

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

      13. pow2 62.3%

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

   8. Applied egg-rr 80.6%

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

      1. pow-plus 80.5%

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

      2. metadata-eval 80.5%

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

      3. associate-/r* 79.8%

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

      4. cube-div 75.1%

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

      5. rem-cube-cbrt 75.1%

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

   10. Simplified 75.1%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999928e224

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-/r* 38.8%

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

      3. sub-neg 38.8%

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

      4. distribute-rgt-in 35.2%

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

      5. unpow2 35.2%

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

      6. times-frac 24.5%

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

      7. sqr-neg 24.5%

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

      8. times-frac 35.2%

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

      9. unpow2 35.2%

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

      10. distribute-rgt-in 38.8%

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

      11. +-commutative 38.8%

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

      12. associate-+l+ 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in t around 0 93.8%

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

      1. associate-*r/ 93.9%

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

      2. associate-*r* 93.9%

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

      3. times-frac 93.9%

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

   7. Simplified 93.9%

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

      1. associate-*l/ 93.9%

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

      2. div-inv 93.8%

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

      3. pow-flip 93.9%

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

      4. metadata-eval 93.9%

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

   9. Applied egg-rr 93.9%

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

       1. times-frac 91.3%

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

   11. Simplified 91.3%

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

       1. frac-times 93.9%

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

       2. associate-*l* 93.9%

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

   13. Applied egg-rr 93.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 10^{+225}\right):\\ \;\;\;\;\frac{2}{\frac{k}{t}} \cdot \frac{\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{\sin k \cdot \tan k}}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{\tan k \cdot \frac{\sin k}{\ell}} \cdot \frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}}{\frac{k \cdot \frac{k}{t\_m}}{t\_m}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t\_m \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{t\_m}} \cdot \frac{\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{\sin k \cdot \tan k}}{\frac{k}{t\_m}}\\ \end{array} \end{array} \]

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 l) 0.0)
    (/
     (/ 2.0 (pow (* (cbrt (* (tan k) (/ (sin k) l))) (/ t_m (cbrt l))) 3.0))
     (/ (* k (/ k t_m)) t_m))
    (if (<= (* l l) 1e+225)
      (/
       (* 2.0 (* (pow l 2.0) (* (cos k) (pow (sin k) -2.0))))
       (* t_m (pow k 2.0)))
      (*
       (/ 2.0 (/ k t_m))
       (/
        (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (* (sin k) (tan k)))
        (/ k t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / pow((cbrt((tan(k) * (sin(k) / l))) * (t_m / cbrt(l))), 3.0)) / ((k * (k / t_m)) / t_m);
	} else if ((l * l) <= 1e+225) {
		tmp = (2.0 * (pow(l, 2.0) * (cos(k) * pow(sin(k), -2.0)))) / (t_m * pow(k, 2.0));
	} else {
		tmp = (2.0 / (k / t_m)) * ((pow((pow(cbrt(l), 2.0) / t_m), 3.0) / (sin(k) * tan(k))) / (k / t_m));
	}
	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 * l) <= 0.0) {
		tmp = (2.0 / Math.pow((Math.cbrt((Math.tan(k) * (Math.sin(k) / l))) * (t_m / Math.cbrt(l))), 3.0)) / ((k * (k / t_m)) / t_m);
	} else if ((l * l) <= 1e+225) {
		tmp = (2.0 * (Math.pow(l, 2.0) * (Math.cos(k) * Math.pow(Math.sin(k), -2.0)))) / (t_m * Math.pow(k, 2.0));
	} else {
		tmp = (2.0 / (k / t_m)) * ((Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / (Math.sin(k) * Math.tan(k))) / (k / t_m));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(sin(k) / l))) * Float64(t_m / cbrt(l))) ^ 3.0)) / Float64(Float64(k * Float64(k / t_m)) / t_m));
	elseif (Float64(l * l) <= 1e+225)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) * (sin(k) ^ -2.0)))) / Float64(t_m * (k ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(k / t_m)) * Float64(Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / Float64(sin(k) * tan(k))) / Float64(k / t_m)));
	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[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+225], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-*l/ 32.4%

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

      4. associate-/l* 32.4%

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

      5. +-commutative 32.4%

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

      6. unpow2 32.4%

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

      7. sqr-neg 32.4%

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

      8. distribute-frac-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. unpow2 32.4%

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

      11. associate--l+ 48.1%

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

      12. metadata-eval 48.1%

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

      13. +-rgt-identity 48.1%

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

      14. unpow2 48.1%

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

      15. distribute-frac-neg 48.1%

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

      16. distribute-frac-neg 48.1%

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

      17. sqr-neg 48.1%

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

      18. unpow2 48.1%

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

   3. Simplified 48.1%

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

      1. associate-/l* 64.3%

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

      2. associate-/r/ 66.4%

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

      3. associate-/l* 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. unpow2 66.3%

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

      2. associate-*r/ 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. add-cube-cbrt 66.2%

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

      2. pow3 66.2%

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

      3. *-commutative 66.2%

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

      4. cbrt-prod 66.1%

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

      5. associate-/r/ 66.2%

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

      6. cbrt-div 66.2%

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

      7. unpow3 66.1%

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

      8. add-cbrt-cube 80.0%

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

   10. Applied egg-rr 80.0%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999928e224

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-/r* 38.8%

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

      3. sub-neg 38.8%

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

      4. distribute-rgt-in 35.2%

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

      5. unpow2 35.2%

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

      6. times-frac 24.5%

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

      7. sqr-neg 24.5%

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

      8. times-frac 35.2%

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

      9. unpow2 35.2%

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

      10. distribute-rgt-in 38.8%

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

      11. +-commutative 38.8%

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

      12. associate-+l+ 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in t around 0 93.8%

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

      1. associate-*r/ 93.9%

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

      2. associate-*r* 93.9%

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

      3. times-frac 93.9%

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

   7. Simplified 93.9%

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

      1. associate-*l/ 93.9%

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

      2. div-inv 93.8%

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

      3. pow-flip 93.9%

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

      4. metadata-eval 93.9%

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

   9. Applied egg-rr 93.9%

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

       1. times-frac 91.3%

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

   11. Simplified 91.3%

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

       1. frac-times 93.9%

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

       2. associate-*l* 93.9%

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

   13. Applied egg-rr 93.9%

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

   if 9.99999999999999928e224 < (*.f64 l l)

   1. Initial program 33.4%

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

      1. associate-*l* 33.4%

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

      2. associate-/r* 33.4%

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

      3. sub-neg 33.4%

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

      4. distribute-rgt-in 27.1%

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

      5. unpow2 27.1%

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

      6. times-frac 20.8%

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

      7. sqr-neg 20.8%

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

      8. times-frac 27.1%

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

      9. unpow2 27.1%

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

      10. distribute-rgt-in 33.4%

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

      11. +-commutative 33.4%

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

      12. associate-+l+ 33.6%

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

   3. Simplified 33.6%

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

      1. +-rgt-identity 33.6%

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

      2. associate-/r* 33.6%

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

      3. unpow2 33.6%

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

      4. associate-/r* 33.6%

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

      5. associate-/l/ 33.6%

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

      6. *-commutative 33.6%

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

      7. associate-/r* 33.6%

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

      8. associate-/r/ 33.6%

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

   6. Applied egg-rr 50.1%

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

      1. add-cube-cbrt 50.0%

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

      2. pow2 50.0%

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

      3. cbrt-div 50.0%

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

      4. unpow2 50.0%

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

      5. cbrt-prod 50.0%

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

      6. pow2 50.0%

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

      7. cbrt-prod 50.0%

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

      8. unpow3 50.0%

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

      9. add-cbrt-cube 50.0%

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

      10. cbrt-div 50.0%

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

      11. unpow2 50.0%

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

      12. cbrt-prod 55.2%

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

      13. pow2 55.2%

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

   8. Applied egg-rr 74.8%

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

      1. pow-plus 74.8%

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

      2. metadata-eval 74.8%

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

      3. associate-/r* 74.8%

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

      4. cube-div 72.3%

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

      5. rem-cube-cbrt 72.3%

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

   10. Simplified 72.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{\tan k \cdot \frac{\sin k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}}{\frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)}{t \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{t}} \cdot \frac{\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{\sin k \cdot \tan k}}{\frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{{t\_m}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}{\frac{k \cdot \frac{k}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {k}^{-2}\right) \cdot \frac{{\sin k}^{-2} \cdot \left({\ell}^{2} \cdot \cos k\right)}{t\_m}\\ \end{array} \end{array} \]

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 l) 0.0)
    (/
     (/ 2.0 (/ (pow (/ (pow t_m 1.5) (/ l (sin k))) 2.0) (cos k)))
     (/ (* k (/ k t_m)) t_m))
    (*
     (* 2.0 (pow k -2.0))
     (/ (* (pow (sin k) -2.0) (* (pow l 2.0) (cos k))) t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / (pow((pow(t_m, 1.5) / (l / sin(k))), 2.0) / cos(k))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = (2.0 * pow(k, -2.0)) * ((pow(sin(k), -2.0) * (pow(l, 2.0) * cos(k))) / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = (2.0d0 / ((((t_m ** 1.5d0) / (l / sin(k))) ** 2.0d0) / cos(k))) / ((k * (k / t_m)) / t_m)
    else
        tmp = (2.0d0 * (k ** (-2.0d0))) * (((sin(k) ** (-2.0d0)) * ((l ** 2.0d0) * cos(k))) / t_m)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / (Math.pow((Math.pow(t_m, 1.5) / (l / Math.sin(k))), 2.0) / Math.cos(k))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = (2.0 * Math.pow(k, -2.0)) * ((Math.pow(Math.sin(k), -2.0) * (Math.pow(l, 2.0) * Math.cos(k))) / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (2.0 / (math.pow((math.pow(t_m, 1.5) / (l / math.sin(k))), 2.0) / math.cos(k))) / ((k * (k / t_m)) / t_m)
	else:
		tmp = (2.0 * math.pow(k, -2.0)) * ((math.pow(math.sin(k), -2.0) * (math.pow(l, 2.0) * math.cos(k))) / t_m)
	return t_s * tmp

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-*l/ 32.4%

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

      4. associate-/l* 32.4%

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

      5. +-commutative 32.4%

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

      6. unpow2 32.4%

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

      7. sqr-neg 32.4%

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

      8. distribute-frac-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. unpow2 32.4%

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

      11. associate--l+ 48.1%

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

      12. metadata-eval 48.1%

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

      13. +-rgt-identity 48.1%

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

      14. unpow2 48.1%

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

      15. distribute-frac-neg 48.1%

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

      16. distribute-frac-neg 48.1%

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

      17. sqr-neg 48.1%

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

      18. unpow2 48.1%

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

   3. Simplified 48.1%

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

      1. associate-/l* 64.3%

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

      2. associate-/r/ 66.4%

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

      3. associate-/l* 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. unpow2 66.3%

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

      2. associate-*r/ 66.3%

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

   8. Applied egg-rr 66.3%

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

   9. Taylor expanded in t around 0 48.0%

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

       1. times-frac 48.1%

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

       2. associate-*r/ 48.1%

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

       3. metadata-eval 48.1%

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

       4. pow-sqr 18.6%

          \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}}{{\ell}^{2}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       5. unpow2 18.6%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{1.5} \cdot {t}^{1.5}}{\color{blue}{\ell \cdot \ell}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       6. times-frac 26.4%

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

       7. unpow2 26.4%

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

       8. swap-sqr 29.2%

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

       9. unpow1 29.2%

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

       10. pow-plus 29.2%

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

       11. associate-/r/ 29.2%

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

       12. metadata-eval 29.2%

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

   11. Simplified 29.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. associate-/r* 36.6%

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

      3. sub-neg 36.6%

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

      4. distribute-rgt-in 31.8%

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

      5. unpow2 31.8%

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

      6. times-frac 22.9%

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

      7. sqr-neg 22.9%

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

      8. times-frac 31.8%

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

      9. unpow2 31.8%

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

      10. distribute-rgt-in 36.6%

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

      11. +-commutative 36.6%

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

      12. associate-+l+ 44.9%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in t around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. associate-*r* 80.0%

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

      3. times-frac 80.0%

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

   7. Simplified 80.0%

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

      1. associate-*l/ 80.0%

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

      2. div-inv 80.0%

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

      3. pow-flip 80.0%

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

      4. metadata-eval 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. times-frac 80.0%

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

   11. Simplified 80.0%

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

       1. expm1-log1p-u 79.4%

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

       2. expm1-udef 58.8%

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

       3. div-inv 58.8%

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

       4. pow-flip 58.8%

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

       5. metadata-eval 58.8%

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

   13. Applied egg-rr 58.8%

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

       1. expm1-def 79.4%

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

       2. expm1-log1p 80.1%

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

   15. Simplified 80.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}{\frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {k}^{-2}\right) \cdot \frac{{\sin k}^{-2} \cdot \left({\ell}^{2} \cdot \cos k\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{{t\_m}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}{\frac{k \cdot \frac{k}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

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 3.3e-175)
    (/
     (/ 2.0 (/ (pow (/ (pow t_m 1.5) (/ l (sin k))) 2.0) (cos k)))
     (/ (* k (/ k t_m)) t_m))
    (*
     2.0
     (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.3e-175) {
		tmp = (2.0 / (pow((pow(t_m, 1.5) / (l / sin(k))), 2.0) / cos(k))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.3d-175) then
        tmp = (2.0d0 / ((((t_m ** 1.5d0) / (l / sin(k))) ** 2.0d0) / cos(k))) / ((k * (k / t_m)) / t_m)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.3e-175) {
		tmp = (2.0 / (Math.pow((Math.pow(t_m, 1.5) / (l / Math.sin(k))), 2.0) / Math.cos(k))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.3e-175:
		tmp = (2.0 / (math.pow((math.pow(t_m, 1.5) / (l / math.sin(k))), 2.0) / math.cos(k))) / ((k * (k / t_m)) / t_m)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.3e-175)
		tmp = Float64(Float64(2.0 / Float64((Float64((t_m ^ 1.5) / Float64(l / sin(k))) ^ 2.0) / cos(k))) / Float64(Float64(k * Float64(k / t_m)) / t_m));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.3e-175)
		tmp = (2.0 / ((((t_m ^ 1.5) / (l / sin(k))) ^ 2.0) / cos(k))) / ((k * (k / t_m)) / t_m);
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 3.3e-175], N[(N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.29999999999999999e-175

   1. Initial program 31.7%

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

      1. associate-/r* 32.1%

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

      2. associate-*l* 32.1%

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

      3. associate-*l/ 32.1%

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

      4. associate-/l* 32.1%

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

      5. +-commutative 32.1%

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

      6. unpow2 32.1%

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

      7. sqr-neg 32.1%

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

      8. distribute-frac-neg 32.1%

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

      9. distribute-frac-neg 32.1%

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

      10. unpow2 32.1%

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

      11. associate--l+ 42.7%

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

      12. metadata-eval 42.7%

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

      13. +-rgt-identity 42.7%

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

      14. unpow2 42.7%

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

      15. distribute-frac-neg 42.7%

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

      16. distribute-frac-neg 42.7%

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

      17. sqr-neg 42.7%

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

      18. unpow2 42.7%

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

   3. Simplified 42.7%

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

      1. associate-/l* 49.7%

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

      2. associate-/r/ 52.0%

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

      3. associate-/l* 52.0%

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

   6. Applied egg-rr 52.0%

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

      1. unpow2 52.0%

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

      2. associate-*r/ 52.0%

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

   8. Applied egg-rr 52.0%

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

   9. Taylor expanded in t around 0 42.7%

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

       1. times-frac 42.7%

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

       2. associate-*r/ 42.7%

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

       3. metadata-eval 42.7%

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

       4. pow-sqr 16.0%

          \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}}{{\ell}^{2}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       5. unpow2 16.0%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{1.5} \cdot {t}^{1.5}}{\color{blue}{\ell \cdot \ell}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       6. times-frac 22.0%

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

       7. unpow2 22.0%

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

       8. swap-sqr 23.2%

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

       9. unpow1 23.2%

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

       10. pow-plus 23.2%

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

       11. associate-/r/ 23.9%

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

       12. metadata-eval 23.9%

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

   11. Simplified 23.9%

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

   if 3.29999999999999999e-175 < l

   1. Initial program 40.5%

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

      1. associate-*l* 40.5%

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

      2. associate-/r* 40.5%

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

      3. sub-neg 40.5%

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

      4. distribute-rgt-in 36.7%

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

      5. unpow2 36.7%

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

      6. times-frac 29.0%

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

      7. sqr-neg 29.0%

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

      8. times-frac 36.7%

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

      9. unpow2 36.7%

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

      10. distribute-rgt-in 40.5%

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

      11. +-commutative 40.5%

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

      12. associate-+l+ 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around 0 83.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}{\frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{{t\_m}^{1.5}}{\frac{\ell}{\tan k}}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 1.9e-93)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k))
      (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
    (if (<= t_m 2.2e+179)
      (/
       (/
        2.0
        (* (/ (pow t_m 1.5) (/ l (sin k))) (/ (pow t_m 1.5) (/ l (tan k)))))
       (* (/ k t_m) (/ k t_m)))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e-93) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 2.2e+179) {
		tmp = (2.0 / ((pow(t_m, 1.5) / (l / sin(k))) * (pow(t_m, 1.5) / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.9d-93) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 2.2d+179) then
        tmp = (2.0d0 / (((t_m ** 1.5d0) / (l / sin(k))) * ((t_m ** 1.5d0) / (l / tan(k))))) / ((k / t_m) * (k / t_m))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e-93) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 2.2e+179) {
		tmp = (2.0 / ((Math.pow(t_m, 1.5) / (l / Math.sin(k))) * (Math.pow(t_m, 1.5) / (l / Math.tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.9e-93:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 2.2e+179:
		tmp = (2.0 / ((math.pow(t_m, 1.5) / (l / math.sin(k))) * (math.pow(t_m, 1.5) / (l / math.tan(k))))) / ((k / t_m) * (k / t_m))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.9e-93)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 2.2e+179)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 1.5) / Float64(l / sin(k))) * Float64((t_m ^ 1.5) / Float64(l / tan(k))))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.9e-93)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 2.2e+179)
		tmp = (2.0 / (((t_m ^ 1.5) / (l / sin(k))) * ((t_m ^ 1.5) / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e-93], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e+179], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-93

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-/r* 35.2%

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

      3. sub-neg 35.2%

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

      4. distribute-rgt-in 30.0%

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

      5. unpow2 30.0%

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

      6. times-frac 20.5%

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

      7. sqr-neg 20.5%

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

      8. times-frac 30.0%

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

      9. unpow2 30.0%

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

      10. distribute-rgt-in 35.2%

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

      11. +-commutative 35.2%

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

      12. associate-+l+ 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in t around 0 78.5%

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

      1. unpow2 78.5%

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

      2. sin-mult 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. div-sub 70.2%

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

      2. +-inverses 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      3. cos-0 70.2%

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

      4. metadata-eval 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      5. count-2 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)\right)} \]

   9. Simplified 70.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)} \]

   if 1.8999999999999999e-93 < t < 2.2e179

   1. Initial program 46.2%

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

      1. associate-/r* 47.1%

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

      2. associate-*l* 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/l* 47.1%

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

      5. +-commutative 47.1%

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

      6. unpow2 47.1%

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

      7. sqr-neg 47.1%

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

      8. distribute-frac-neg 47.1%

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

      9. distribute-frac-neg 47.1%

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

      10. unpow2 47.1%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 57.9%

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

      2. times-frac 64.0%

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

      3. times-frac 68.2%

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

      4. sqrt-pow1 68.2%

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

      5. metadata-eval 68.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\frac{\ell}{\sin k}} \cdot \frac{\sqrt{{t}^{3}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 84.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. unpow2 76.9%

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

   8. Applied egg-rr 84.1%

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

   if 2.2e179 < t

   1. Initial program 9.1%

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

      1. associate-*l* 9.1%

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

      2. associate-/r* 9.1%

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

      3. sub-neg 9.1%

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

      4. distribute-rgt-in 9.1%

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

      5. unpow2 9.1%

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

      6. times-frac 0.0%

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

      7. sqr-neg 0.0%

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

      8. times-frac 9.1%

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

      9. unpow2 9.1%

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

      10. distribute-rgt-in 9.1%

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

      11. +-commutative 9.1%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 77.9%

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

      3. times-frac 77.9%

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

   7. Simplified 77.9%

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

      1. associate-*l/ 77.9%

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

      2. div-inv 77.8%

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

      3. pow-flip 77.9%

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

      4. metadata-eval 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. times-frac 82.2%

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

   11. Simplified 82.2%

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

   12. Taylor expanded in k around 0 69.0%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 69.0%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 73.6%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 73.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 73.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 73.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 73.5%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{{t\_m}^{1.5}}{\frac{\ell}{\tan k}}}}{\frac{k}{t\_m \cdot \frac{t\_m}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 6.5e-94)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k))
      (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
    (if (<= t_m 6.1e+178)
      (/
       (/
        2.0
        (* (/ (pow t_m 1.5) (/ l (sin k))) (/ (pow t_m 1.5) (/ l (tan k)))))
       (/ k (* t_m (/ t_m k))))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-94) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 6.1e+178) {
		tmp = (2.0 / ((pow(t_m, 1.5) / (l / sin(k))) * (pow(t_m, 1.5) / (l / tan(k))))) / (k / (t_m * (t_m / k)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6.5d-94) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 6.1d+178) then
        tmp = (2.0d0 / (((t_m ** 1.5d0) / (l / sin(k))) * ((t_m ** 1.5d0) / (l / tan(k))))) / (k / (t_m * (t_m / k)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-94) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 6.1e+178) {
		tmp = (2.0 / ((Math.pow(t_m, 1.5) / (l / Math.sin(k))) * (Math.pow(t_m, 1.5) / (l / Math.tan(k))))) / (k / (t_m * (t_m / k)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6.5e-94:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 6.1e+178:
		tmp = (2.0 / ((math.pow(t_m, 1.5) / (l / math.sin(k))) * (math.pow(t_m, 1.5) / (l / math.tan(k))))) / (k / (t_m * (t_m / k)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.5e-94)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 6.1e+178)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 1.5) / Float64(l / sin(k))) * Float64((t_m ^ 1.5) / Float64(l / tan(k))))) / Float64(k / Float64(t_m * Float64(t_m / k))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6.5e-94)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 6.1e+178)
		tmp = (2.0 / (((t_m ^ 1.5) / (l / sin(k))) * ((t_m ^ 1.5) / (l / tan(k))))) / (k / (t_m * (t_m / k)));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-94], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e+178], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.4999999999999996e-94

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-/r* 35.2%

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

      3. sub-neg 35.2%

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

      4. distribute-rgt-in 30.0%

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

      5. unpow2 30.0%

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

      6. times-frac 20.5%

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

      7. sqr-neg 20.5%

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

      8. times-frac 30.0%

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

      9. unpow2 30.0%

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

      10. distribute-rgt-in 35.2%

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

      11. +-commutative 35.2%

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

      12. associate-+l+ 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in t around 0 78.5%

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

      1. unpow2 78.5%

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

      2. sin-mult 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. div-sub 70.2%

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

      2. +-inverses 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      3. cos-0 70.2%

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

      4. metadata-eval 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      5. count-2 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)\right)} \]

   9. Simplified 70.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)} \]

   if 6.4999999999999996e-94 < t < 6.1000000000000001e178

   1. Initial program 46.2%

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

      1. associate-/r* 47.1%

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

      2. associate-*l* 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/l* 47.1%

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

      5. +-commutative 47.1%

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

      6. unpow2 47.1%

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

      7. sqr-neg 47.1%

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

      8. distribute-frac-neg 47.1%

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

      9. distribute-frac-neg 47.1%

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

      10. unpow2 47.1%

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

      11. associate--l+ 57.9%

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

      12. metadata-eval 57.9%

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

      13. +-rgt-identity 57.9%

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

      14. unpow2 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. distribute-frac-neg 57.9%

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

      17. sqr-neg 57.9%

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

      18. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 57.9%

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

      2. times-frac 64.0%

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

      3. times-frac 68.2%

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

      4. sqrt-pow1 68.2%

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

      5. metadata-eval 68.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\frac{\ell}{\sin k}} \cdot \frac{\sqrt{{t}^{3}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 84.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. unpow2 84.1%

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

      2. clear-num 84.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t}} \]

      3. frac-times 84.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{1 \cdot k}{\frac{t}{k} \cdot t}}} \]

      4. *-un-lft-identity 84.1%

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

   8. Applied egg-rr 84.1%

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

   if 6.1000000000000001e178 < t

   1. Initial program 9.1%

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

      1. associate-*l* 9.1%

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

      2. associate-/r* 9.1%

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

      3. sub-neg 9.1%

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

      4. distribute-rgt-in 9.1%

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

      5. unpow2 9.1%

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

      6. times-frac 0.0%

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

      7. sqr-neg 0.0%

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

      8. times-frac 9.1%

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

      9. unpow2 9.1%

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

      10. distribute-rgt-in 9.1%

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

      11. +-commutative 9.1%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 77.9%

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

      3. times-frac 77.9%

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

   7. Simplified 77.9%

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

      1. associate-*l/ 77.9%

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

      2. div-inv 77.8%

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

      3. pow-flip 77.9%

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

      4. metadata-eval 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. times-frac 82.2%

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

   11. Simplified 82.2%

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

   12. Taylor expanded in k around 0 69.0%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 69.0%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 73.6%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 73.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 73.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 73.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 73.5%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{1.5}}{\frac{\ell}{\sin k}} \cdot \frac{{t}^{1.5}}{\frac{\ell}{\tan k}}}}{\frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{{t\_m}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}{\frac{k \cdot \frac{k}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 4.6e-93)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k))
      (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
    (if (<= t_m 1.25e+179)
      (/
       (/ 2.0 (/ (pow (/ (pow t_m 1.5) (/ l (sin k))) 2.0) (cos k)))
       (/ (* k (/ k t_m)) t_m))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.6e-93) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 1.25e+179) {
		tmp = (2.0 / (pow((pow(t_m, 1.5) / (l / sin(k))), 2.0) / cos(k))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.6d-93) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 1.25d+179) then
        tmp = (2.0d0 / ((((t_m ** 1.5d0) / (l / sin(k))) ** 2.0d0) / cos(k))) / ((k * (k / t_m)) / t_m)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.6e-93) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 1.25e+179) {
		tmp = (2.0 / (Math.pow((Math.pow(t_m, 1.5) / (l / Math.sin(k))), 2.0) / Math.cos(k))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.6e-93:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 1.25e+179:
		tmp = (2.0 / (math.pow((math.pow(t_m, 1.5) / (l / math.sin(k))), 2.0) / math.cos(k))) / ((k * (k / t_m)) / t_m)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.6e-93)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 1.25e+179)
		tmp = Float64(Float64(2.0 / Float64((Float64((t_m ^ 1.5) / Float64(l / sin(k))) ^ 2.0) / cos(k))) / Float64(Float64(k * Float64(k / t_m)) / t_m));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.6e-93)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 1.25e+179)
		tmp = (2.0 / ((((t_m ^ 1.5) / (l / sin(k))) ^ 2.0) / cos(k))) / ((k * (k / t_m)) / t_m);
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.6e-93], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+179], N[(N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.5999999999999996e-93

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-/r* 35.2%

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

      3. sub-neg 35.2%

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

      4. distribute-rgt-in 30.0%

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

      5. unpow2 30.0%

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

      6. times-frac 20.5%

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

      7. sqr-neg 20.5%

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

      8. times-frac 30.0%

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

      9. unpow2 30.0%

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

      10. distribute-rgt-in 35.2%

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

      11. +-commutative 35.2%

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

      12. associate-+l+ 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in t around 0 78.5%

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

      1. unpow2 78.5%

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

      2. sin-mult 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. div-sub 70.2%

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

      2. +-inverses 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      3. cos-0 70.2%

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

      4. metadata-eval 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      5. count-2 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)\right)} \]

   9. Simplified 70.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)} \]

   if 4.5999999999999996e-93 < t < 1.25e179

   1. Initial program 46.2%

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

      1. associate-/r* 47.1%

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

      2. associate-*l* 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/l* 47.1%

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

      5. +-commutative 47.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 47.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 47.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 47.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 47.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 47.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 57.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 64.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k \cdot \tan k}{\ell}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 68.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l* 68.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 68.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 68.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      2. associate-*r/ 68.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}} \]

   8. Applied egg-rr 68.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}} \]

   9. Taylor expanded in t around 0 57.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}}}{\frac{\frac{k}{t} \cdot k}{t}} \]
   10. Step-by-step derivation

       1. times-frac 57.8%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       2. associate-*r/ 57.8%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{{\ell}^{2}} \cdot {\sin k}^{2}}{\cos k}}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       3. metadata-eval 57.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{\color{blue}{\left(2 \cdot 1.5\right)}}}{{\ell}^{2}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       4. pow-sqr 57.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}}{{\ell}^{2}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       5. unpow2 57.8%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{1.5} \cdot {t}^{1.5}}{\color{blue}{\ell \cdot \ell}} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       6. times-frac 73.5%

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot {\sin k}^{2}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       7. unpow2 73.5%

          \[\leadsto \frac{\frac{2}{\frac{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       8. swap-sqr 78.8%

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sin k\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sin k\right)}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       9. unpow1 78.8%

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sin k\right)}^{1}} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sin k\right)}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       10. pow-plus 78.8%

           \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sin k\right)}^{\left(1 + 1\right)}}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       11. associate-/r/ 80.5%

           \[\leadsto \frac{\frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}}^{\left(1 + 1\right)}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

       12. metadata-eval 80.5%

           \[\leadsto \frac{\frac{2}{\frac{{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{\color{blue}{2}}}{\cos k}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

   11. Simplified 80.5%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

   if 1.25e179 < t

   1. Initial program 9.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 9.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 9.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 9.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 9.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 9.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 9.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 9.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 9.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 9.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 36.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 78.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 77.9%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 77.9%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 77.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 77.8%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 77.9%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 77.9%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 82.2%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 82.2%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 69.0%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 69.0%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 69.0%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 73.6%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 73.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 73.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 73.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 73.5%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k}}}{\frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 4e-93)
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k))
      (* (pow k 2.0) (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
    (if (<= t_m 1.9e+162)
      (/
       (/ 2.0 (* (* (sin k) (/ (pow t_m 2.0) l)) (/ t_m (/ l (tan k)))))
       (* (/ k t_m) (/ k t_m)))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-93) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 1.9e+162) {
		tmp = (2.0 / ((sin(k) * (pow(t_m, 2.0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4d-93) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    else if (t_m <= 1.9d+162) then
        tmp = (2.0d0 / ((sin(k) * ((t_m ** 2.0d0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-93) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t_m <= 1.9e+162) {
		tmp = (2.0 / ((Math.sin(k) * (Math.pow(t_m, 2.0) / l)) * (t_m / (l / Math.tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4e-93:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t_m * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t_m <= 1.9e+162:
		tmp = (2.0 / ((math.sin(k) * (math.pow(t_m, 2.0) / l)) * (t_m / (l / math.tan(k))))) / ((k / t_m) * (k / t_m))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e-93)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t_m <= 1.9e+162)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 2.0) / l)) * Float64(t_m / Float64(l / tan(k))))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4e-93)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t_m <= 1.9e+162)
		tmp = (2.0 / ((sin(k) * ((t_m ^ 2.0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-93], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+162], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.9999999999999996e-93

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 30.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 30.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 20.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 20.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 30.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 30.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.2%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.2%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 78.5%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}\right)} \]

      2. sin-mult 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}\right)} \]

   7. Applied egg-rr 70.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}\right)} \]
   8. Step-by-step derivation

      1. div-sub 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(\frac{\cos \left(k - k\right)}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right)} \]

      2. +-inverses 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      3. cos-0 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      4. metadata-eval 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)\right)} \]

      5. count-2 70.2%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)\right)} \]

   9. Simplified 70.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)} \]

   if 3.9999999999999996e-93 < t < 1.90000000000000012e162

   1. Initial program 51.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 52.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 52.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 52.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 52.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 52.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 52.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 52.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 52.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 52.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 52.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 60.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 60.8%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 67.6%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 80.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 80.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 80.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 80.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 80.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 80.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \sin k\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   10. Applied egg-rr 80.5%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \sin k\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 1.90000000000000012e162 < t

   1. Initial program 7.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 7.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 7.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 7.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 7.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 36.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 75.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 75.5%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 75.5%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 75.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 75.4%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 75.5%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 75.5%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 78.8%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 68.5%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 68.5%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 72.1%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 72.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 72.1%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 72.1%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 72.1%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 8.4e-166)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow k 2.0)))))
    (if (<= t_m 1.9e+162)
      (/
       (/ 2.0 (* (* (sin k) (/ (pow t_m 2.0) l)) (/ t_m (/ l (tan k)))))
       (* (/ k t_m) (/ k t_m)))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(k, 2.0))));
	} else if (t_m <= 1.9e+162) {
		tmp = (2.0 / ((sin(k) * (pow(t_m, 2.0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m * (k ** 2.0d0))))
    else if (t_m <= 1.9d+162) then
        tmp = (2.0d0 / ((sin(k) * ((t_m ** 2.0d0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(k, 2.0))));
	} else if (t_m <= 1.9e+162) {
		tmp = (2.0 / ((Math.sin(k) * (Math.pow(t_m, 2.0) / l)) * (t_m / (l / Math.tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t_m * math.pow(k, 2.0))))
	elif t_m <= 1.9e+162:
		tmp = (2.0 / ((math.sin(k) * (math.pow(t_m, 2.0) / l)) * (t_m / (l / math.tan(k))))) / ((k / t_m) * (k / t_m))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (k ^ 2.0)))));
	elseif (t_m <= 1.9e+162)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 2.0) / l)) * Float64(t_m / Float64(l / tan(k))))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m * (k ^ 2.0))));
	elseif (t_m <= 1.9e+162)
		tmp = (2.0 / ((sin(k) * ((t_m ^ 2.0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+162], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 69.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   if 8.3999999999999998e-166 < t < 1.90000000000000012e162

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 49.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 49.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 55.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 63.9%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 77.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 77.7%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 77.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 77.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 77.7%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 77.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \sin k\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   10. Applied egg-rr 77.8%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \sin k\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 1.90000000000000012e162 < t

   1. Initial program 7.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 7.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 7.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 7.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 7.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 36.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 75.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 75.5%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 75.5%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 75.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 75.4%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 75.5%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 75.5%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 78.8%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 68.5%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 68.5%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 72.1%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 72.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 72.1%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 72.1%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 72.1%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\ell}^{2}}{{k}^{2}}\\ t_3 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{t\_3} \cdot \left({t\_m}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{-103}:\\ \;\;\;\;t\_2 \cdot \frac{2}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k}{t\_3}}}{\frac{k \cdot \frac{k}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 l 2.0) (pow k 2.0))) (t_3 (/ l (tan k))))
   (*
    t_s
    (if (<= t_m 8.4e-166)
      (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
      (if (<= t_m 2.1e-121)
        (/
         (/ 2.0 (* (/ t_m t_3) (* (pow t_m 2.0) (/ k l))))
         (* (/ k t_m) (/ k t_m)))
        (if (<= t_m 1.9e-103)
          (* t_2 (/ 2.0 (* t_m (pow k 2.0))))
          (if (<= t_m 4.5e+78)
            (/
             (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (sin k) t_3)))
             (/ (* k (/ k t_m)) t_m))
            (*
             2.0
             (*
              t_2
              (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) / pow(k, 2.0);
	double t_3 = l / tan(k);
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 2.1e-121) {
		tmp = (2.0 / ((t_m / t_3) * (pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	} else if (t_m <= 1.9e-103) {
		tmp = t_2 * (2.0 / (t_m * pow(k, 2.0)));
	} else if (t_m <= 4.5e+78) {
		tmp = (2.0 / ((pow(t_m, 3.0) / l) * (sin(k) / t_3))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * (t_2 * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (l ** 2.0d0) / (k ** 2.0d0)
    t_3 = l / tan(k)
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 2.1d-121) then
        tmp = (2.0d0 / ((t_m / t_3) * ((t_m ** 2.0d0) * (k / l)))) / ((k / t_m) * (k / t_m))
    else if (t_m <= 1.9d-103) then
        tmp = t_2 * (2.0d0 / (t_m * (k ** 2.0d0)))
    else if (t_m <= 4.5d+78) then
        tmp = (2.0d0 / (((t_m ** 3.0d0) / l) * (sin(k) / t_3))) / ((k * (k / t_m)) / t_m)
    else
        tmp = 2.0d0 * (t_2 * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(l, 2.0) / Math.pow(k, 2.0);
	double t_3 = l / Math.tan(k);
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 2.1e-121) {
		tmp = (2.0 / ((t_m / t_3) * (Math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	} else if (t_m <= 1.9e-103) {
		tmp = t_2 * (2.0 / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 4.5e+78) {
		tmp = (2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) / t_3))) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * (t_2 * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) / math.pow(k, 2.0)
	t_3 = l / math.tan(k)
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 2.1e-121:
		tmp = (2.0 / ((t_m / t_3) * (math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m))
	elif t_m <= 1.9e-103:
		tmp = t_2 * (2.0 / (t_m * math.pow(k, 2.0)))
	elif t_m <= 4.5e+78:
		tmp = (2.0 / ((math.pow(t_m, 3.0) / l) * (math.sin(k) / t_3))) / ((k * (k / t_m)) / t_m)
	else:
		tmp = 2.0 * (t_2 * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) / (k ^ 2.0))
	t_3 = Float64(l / tan(k))
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 2.1e-121)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / t_3) * Float64((t_m ^ 2.0) * Float64(k / l)))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	elseif (t_m <= 1.9e-103)
		tmp = Float64(t_2 * Float64(2.0 / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 4.5e+78)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) / t_3))) / Float64(Float64(k * Float64(k / t_m)) / t_m));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) / (k ^ 2.0);
	t_3 = l / tan(k);
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 2.1e-121)
		tmp = (2.0 / ((t_m / t_3) * ((t_m ^ 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	elseif (t_m <= 1.9e-103)
		tmp = t_2 * (2.0 / (t_m * (k ^ 2.0)));
	elseif (t_m <= 4.5e+78)
		tmp = (2.0 / (((t_m ^ 3.0) / l) * (sin(k) / t_3))) / ((k * (k / t_m)) / t_m);
	else
		tmp = 2.0 * (t_2 * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e-121], N[(N[(2.0 / N[(N[(t$95$m / t$95$3), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e-103], N[(t$95$2 * N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+78], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 2.0999999999999999e-121

   1. Initial program 33.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 33.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 33.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 50.7%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 71.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 71.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 71.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 71.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 71.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
   10. Step-by-step derivation

       1. associate-/l* 71.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

       2. associate-/r/ 71.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   11. Simplified 71.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 2.0999999999999999e-121 < t < 1.9e-103

   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. associate-/r* 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 100.0%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 100.0%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 100.0%

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]

   if 1.9e-103 < t < 4.4999999999999999e78

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 60.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 74.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k \cdot \tan k}{\ell}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 78.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l* 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 78.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      2. associate-*r/ 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}} \]

   8. Applied egg-rr 78.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}} \]

   if 4.4999999999999999e78 < t

   1. Initial program 12.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 12.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 12.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 12.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 7.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 7.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 12.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 12.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 38.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 77.2%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 77.2%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 77.1%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 77.2%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 77.2%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 79.6%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 79.6%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 62.4%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 62.4%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 70.2%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 70.2%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 70.2%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 70.2%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 70.2%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \left({t}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\ell}^{2}}{{k}^{2}}\\ t_3 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{t\_3} \cdot \left({t\_m}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{-103}:\\ \;\;\;\;t\_2 \cdot \frac{2}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k}{t\_3}}}{\frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 l 2.0) (pow k 2.0))) (t_3 (/ l (tan k))))
   (*
    t_s
    (if (<= t_m 8.4e-166)
      (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
      (if (<= t_m 2.5e-121)
        (/
         (/ 2.0 (* (/ t_m t_3) (* (pow t_m 2.0) (/ k l))))
         (* (/ k t_m) (/ k t_m)))
        (if (<= t_m 4.5e-103)
          (* t_2 (/ 2.0 (* t_m (pow k 2.0))))
          (if (<= t_m 2.1e+79)
            (/
             (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (sin k) t_3)))
             (/ (/ k t_m) (/ t_m k)))
            (*
             2.0
             (*
              t_2
              (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) / pow(k, 2.0);
	double t_3 = l / tan(k);
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 2.5e-121) {
		tmp = (2.0 / ((t_m / t_3) * (pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	} else if (t_m <= 4.5e-103) {
		tmp = t_2 * (2.0 / (t_m * pow(k, 2.0)));
	} else if (t_m <= 2.1e+79) {
		tmp = (2.0 / ((pow(t_m, 3.0) / l) * (sin(k) / t_3))) / ((k / t_m) / (t_m / k));
	} else {
		tmp = 2.0 * (t_2 * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (l ** 2.0d0) / (k ** 2.0d0)
    t_3 = l / tan(k)
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 2.5d-121) then
        tmp = (2.0d0 / ((t_m / t_3) * ((t_m ** 2.0d0) * (k / l)))) / ((k / t_m) * (k / t_m))
    else if (t_m <= 4.5d-103) then
        tmp = t_2 * (2.0d0 / (t_m * (k ** 2.0d0)))
    else if (t_m <= 2.1d+79) then
        tmp = (2.0d0 / (((t_m ** 3.0d0) / l) * (sin(k) / t_3))) / ((k / t_m) / (t_m / k))
    else
        tmp = 2.0d0 * (t_2 * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(l, 2.0) / Math.pow(k, 2.0);
	double t_3 = l / Math.tan(k);
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 2.5e-121) {
		tmp = (2.0 / ((t_m / t_3) * (Math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	} else if (t_m <= 4.5e-103) {
		tmp = t_2 * (2.0 / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 2.1e+79) {
		tmp = (2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) / t_3))) / ((k / t_m) / (t_m / k));
	} else {
		tmp = 2.0 * (t_2 * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) / math.pow(k, 2.0)
	t_3 = l / math.tan(k)
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 2.5e-121:
		tmp = (2.0 / ((t_m / t_3) * (math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m))
	elif t_m <= 4.5e-103:
		tmp = t_2 * (2.0 / (t_m * math.pow(k, 2.0)))
	elif t_m <= 2.1e+79:
		tmp = (2.0 / ((math.pow(t_m, 3.0) / l) * (math.sin(k) / t_3))) / ((k / t_m) / (t_m / k))
	else:
		tmp = 2.0 * (t_2 * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) / (k ^ 2.0))
	t_3 = Float64(l / tan(k))
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 2.5e-121)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / t_3) * Float64((t_m ^ 2.0) * Float64(k / l)))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	elseif (t_m <= 4.5e-103)
		tmp = Float64(t_2 * Float64(2.0 / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 2.1e+79)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) / t_3))) / Float64(Float64(k / t_m) / Float64(t_m / k)));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) / (k ^ 2.0);
	t_3 = l / tan(k);
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 2.5e-121)
		tmp = (2.0 / ((t_m / t_3) * ((t_m ^ 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	elseif (t_m <= 4.5e-103)
		tmp = t_2 * (2.0 / (t_m * (k ^ 2.0)));
	elseif (t_m <= 2.1e+79)
		tmp = (2.0 / (((t_m ^ 3.0) / l) * (sin(k) / t_3))) / ((k / t_m) / (t_m / k));
	else
		tmp = 2.0 * (t_2 * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e-121], N[(N[(2.0 / N[(N[(t$95$m / t$95$3), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e-103], N[(t$95$2 * N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+79], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 2.49999999999999995e-121

   1. Initial program 33.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 33.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 33.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 50.7%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 71.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 71.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 71.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 71.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 71.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
   10. Step-by-step derivation

       1. associate-/l* 71.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

       2. associate-/r/ 71.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   11. Simplified 71.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 2.49999999999999995e-121 < t < 4.5e-103

   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. associate-/r* 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 100.0%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 100.0%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 100.0%

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]

   if 4.5e-103 < t < 2.10000000000000008e79

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 60.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 74.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k \cdot \tan k}{\ell}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 78.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l* 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 78.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      2. clear-num 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}} \]

      3. un-div-inv 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}} \]

   8. Applied egg-rr 78.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}} \]

   if 2.10000000000000008e79 < t

   1. Initial program 12.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 12.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 12.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 12.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 7.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 7.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 12.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 12.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 38.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 77.2%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 77.2%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 77.1%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 77.2%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 77.2%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 79.6%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 79.6%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 62.4%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 62.4%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 70.2%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 70.2%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 70.2%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 70.2%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 70.2%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \left({t}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\ell}^{2}}{{k}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\frac{\ell}{\tan k}} \cdot \left({t\_m}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{-103}:\\ \;\;\;\;t\_2 \cdot \frac{2}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot {t\_m}^{3}}{\ell}}}{\frac{k \cdot \frac{k}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 l 2.0) (pow k 2.0))))
   (*
    t_s
    (if (<= t_m 8.4e-166)
      (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
      (if (<= t_m 2.3e-121)
        (/
         (/ 2.0 (* (/ t_m (/ l (tan k))) (* (pow t_m 2.0) (/ k l))))
         (* (/ k t_m) (/ k t_m)))
        (if (<= t_m 3.9e-103)
          (* t_2 (/ 2.0 (* t_m (pow k 2.0))))
          (if (<= t_m 4.2e+79)
            (/
             (/ 2.0 (/ (* (* (tan k) (/ (sin k) l)) (pow t_m 3.0)) l))
             (/ (* k (/ k t_m)) t_m))
            (*
             2.0
             (*
              t_2
              (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(l, 2.0) / pow(k, 2.0);
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 2.3e-121) {
		tmp = (2.0 / ((t_m / (l / tan(k))) * (pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	} else if (t_m <= 3.9e-103) {
		tmp = t_2 * (2.0 / (t_m * pow(k, 2.0)));
	} else if (t_m <= 4.2e+79) {
		tmp = (2.0 / (((tan(k) * (sin(k) / l)) * pow(t_m, 3.0)) / l)) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * (t_2 * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l ** 2.0d0) / (k ** 2.0d0)
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 2.3d-121) then
        tmp = (2.0d0 / ((t_m / (l / tan(k))) * ((t_m ** 2.0d0) * (k / l)))) / ((k / t_m) * (k / t_m))
    else if (t_m <= 3.9d-103) then
        tmp = t_2 * (2.0d0 / (t_m * (k ** 2.0d0)))
    else if (t_m <= 4.2d+79) then
        tmp = (2.0d0 / (((tan(k) * (sin(k) / l)) * (t_m ** 3.0d0)) / l)) / ((k * (k / t_m)) / t_m)
    else
        tmp = 2.0d0 * (t_2 * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(l, 2.0) / Math.pow(k, 2.0);
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 2.3e-121) {
		tmp = (2.0 / ((t_m / (l / Math.tan(k))) * (Math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	} else if (t_m <= 3.9e-103) {
		tmp = t_2 * (2.0 / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 4.2e+79) {
		tmp = (2.0 / (((Math.tan(k) * (Math.sin(k) / l)) * Math.pow(t_m, 3.0)) / l)) / ((k * (k / t_m)) / t_m);
	} else {
		tmp = 2.0 * (t_2 * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(l, 2.0) / math.pow(k, 2.0)
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 2.3e-121:
		tmp = (2.0 / ((t_m / (l / math.tan(k))) * (math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m))
	elif t_m <= 3.9e-103:
		tmp = t_2 * (2.0 / (t_m * math.pow(k, 2.0)))
	elif t_m <= 4.2e+79:
		tmp = (2.0 / (((math.tan(k) * (math.sin(k) / l)) * math.pow(t_m, 3.0)) / l)) / ((k * (k / t_m)) / t_m)
	else:
		tmp = 2.0 * (t_2 * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((l ^ 2.0) / (k ^ 2.0))
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 2.3e-121)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / Float64(l / tan(k))) * Float64((t_m ^ 2.0) * Float64(k / l)))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	elseif (t_m <= 3.9e-103)
		tmp = Float64(t_2 * Float64(2.0 / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 4.2e+79)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(sin(k) / l)) * (t_m ^ 3.0)) / l)) / Float64(Float64(k * Float64(k / t_m)) / t_m));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l ^ 2.0) / (k ^ 2.0);
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 2.3e-121)
		tmp = (2.0 / ((t_m / (l / tan(k))) * ((t_m ^ 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	elseif (t_m <= 3.9e-103)
		tmp = t_2 * (2.0 / (t_m * (k ^ 2.0)));
	elseif (t_m <= 4.2e+79)
		tmp = (2.0 / (((tan(k) * (sin(k) / l)) * (t_m ^ 3.0)) / l)) / ((k * (k / t_m)) / t_m);
	else
		tmp = 2.0 * (t_2 * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e-121], N[(N[(2.0 / N[(N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.9e-103], N[(t$95$2 * N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e+79], N[(N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 2.30000000000000012e-121

   1. Initial program 33.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 33.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 33.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 33.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 33.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 33.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 50.7%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 71.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 71.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 71.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 71.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 71.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
   10. Step-by-step derivation

       1. associate-/l* 71.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

       2. associate-/r/ 71.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   11. Simplified 71.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 2.30000000000000012e-121 < t < 3.9000000000000002e-103

   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. associate-/r* 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 0.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 100.0%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 100.0%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 100.0%

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]

   if 3.9000000000000002e-103 < t < 4.20000000000000016e79

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 60.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 60.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 65.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 74.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k \cdot \tan k}{\ell}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 78.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l* 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 78.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      2. associate-*r/ 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}} \]

   8. Applied egg-rr 78.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{\frac{k}{t} \cdot k}{t}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 78.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \frac{\sin k}{\frac{\ell}{\tan k}}}{\ell}}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

      2. associate-/r/ 78.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell} \cdot \tan k\right)}}{\ell}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

   10. Applied egg-rr 78.0%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)}{\ell}}}}{\frac{\frac{k}{t} \cdot k}{t}} \]

   if 4.20000000000000016e79 < t

   1. Initial program 12.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 12.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 12.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 12.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 7.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 7.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 12.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 12.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 12.9%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 38.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 77.2%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 77.2%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 77.1%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 77.2%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 77.2%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 79.6%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 79.6%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 62.4%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 62.4%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 62.4%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 70.2%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 70.2%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 70.2%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 70.2%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 70.2%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \left({t}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot {t}^{3}}{\ell}}}{\frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{t\_m}{\frac{\ell}{\tan k}}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 8.4e-166)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
    (if (<= t_m 1.9e+162)
      (/
       (/ 2.0 (* (* (sin k) (/ (pow t_m 2.0) l)) (/ t_m (/ l (tan k)))))
       (* (/ k t_m) (/ k t_m)))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 1.9e+162) {
		tmp = (2.0 / ((sin(k) * (pow(t_m, 2.0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 1.9d+162) then
        tmp = (2.0d0 / ((sin(k) * ((t_m ** 2.0d0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 1.9e+162) {
		tmp = (2.0 / ((Math.sin(k) * (Math.pow(t_m, 2.0) / l)) * (t_m / (l / Math.tan(k))))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 1.9e+162:
		tmp = (2.0 / ((math.sin(k) * (math.pow(t_m, 2.0) / l)) * (t_m / (l / math.tan(k))))) / ((k / t_m) * (k / t_m))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 1.9e+162)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 2.0) / l)) * Float64(t_m / Float64(l / tan(k))))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 1.9e+162)
		tmp = (2.0 / ((sin(k) * ((t_m ^ 2.0) / l)) * (t_m / (l / tan(k))))) / ((k / t_m) * (k / t_m));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+162], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 1.90000000000000012e162

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 49.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 49.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 55.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 55.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 63.9%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 77.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 77.7%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 77.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 77.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 77.7%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 77.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \sin k\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   10. Applied egg-rr 77.8%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \sin k\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 1.90000000000000012e162 < t

   1. Initial program 7.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 7.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 7.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 7.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 7.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 7.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 36.0%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 75.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 75.5%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 75.5%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 75.5%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 75.4%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 75.5%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 75.5%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 78.8%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 68.5%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 68.5%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 68.5%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 72.1%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 72.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 72.1%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 72.1%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 72.1%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m \cdot k}{\ell}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t\_m}\right)\\ \end{array} \end{array} \]

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 8.4e-166)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
    (if (<= t_m 2.7e+57)
      (/
       (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ (* t_m k) l)))
       (* (/ k t_m) (/ k t_m)))
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k 2.0))
        (/ (+ (/ 1.0 (pow k 2.0)) -0.16666666666666666) t_m)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 2.7e+57) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (((1.0 / pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 2.7d+57) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (((1.0d0 / (k ** 2.0d0)) + (-0.16666666666666666d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 2.7e+57) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (((1.0 / Math.pow(k, 2.0)) + -0.16666666666666666) / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 2.7e+57:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (((1.0 / math.pow(k, 2.0)) + -0.16666666666666666) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 2.7e+57)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(Float64(t_m * k) / l))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 2.7e+57)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (((1.0 / (k ^ 2.0)) + -0.16666666666666666) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e+57], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 2.6999999999999998e57

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 53.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 55.8%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 67.1%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 73.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 73.0%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 73.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 73.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 73.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 2.6999999999999998e57 < t

   1. Initial program 16.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 16.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 16.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 11.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 11.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 16.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 16.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 42.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 42.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 77.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 77.0%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 76.9%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 76.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 76.9%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 77.0%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 77.0%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 79.2%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 61.3%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 61.3%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

   15. Taylor expanded in l around 0 70.6%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{{k}^{2} \cdot t}} \]
   16. Step-by-step derivation

       1. times-frac 70.6%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       2. sub-neg 70.6%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{t}\right) \]

       3. metadata-eval 70.6%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + \color{blue}{-0.16666666666666666}}{t}\right) \]

   17. Simplified 70.6%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot k}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} + -0.16666666666666666}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m \cdot k}{\ell}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {k}^{-2}}{t\_m} \cdot \left({\ell}^{2} \cdot \left({k}^{-2} + -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

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 8.4e-166)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
    (if (<= t_m 5.2e+57)
      (/
       (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ (* t_m k) l)))
       (* (/ k t_m) (/ k t_m)))
      (*
       (/ (* 2.0 (pow k -2.0)) t_m)
       (* (pow l 2.0) (+ (pow k -2.0) -0.16666666666666666)))))))

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 <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 5.2e+57) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = ((2.0 * pow(k, -2.0)) / t_m) * (pow(l, 2.0) * (pow(k, -2.0) + -0.16666666666666666));
	}
	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 <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 5.2d+57) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
    else
        tmp = ((2.0d0 * (k ** (-2.0d0))) / t_m) * ((l ** 2.0d0) * ((k ** (-2.0d0)) + (-0.16666666666666666d0)))
    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 <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 5.2e+57) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = ((2.0 * Math.pow(k, -2.0)) / t_m) * (Math.pow(l, 2.0) * (Math.pow(k, -2.0) + -0.16666666666666666));
	}
	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 <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 5.2e+57:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
	else:
		tmp = ((2.0 * math.pow(k, -2.0)) / t_m) * (math.pow(l, 2.0) * (math.pow(k, -2.0) + -0.16666666666666666))
	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 <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 5.2e+57)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(Float64(t_m * k) / l))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * (k ^ -2.0)) / t_m) * Float64((l ^ 2.0) * Float64((k ^ -2.0) + -0.16666666666666666)));
	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 <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 5.2e+57)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	else
		tmp = ((2.0 * (k ^ -2.0)) / t_m) * ((l ^ 2.0) * ((k ^ -2.0) + -0.16666666666666666));
	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, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e+57], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k, -2.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 5.2e57

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 53.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 55.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 55.8%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 67.1%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 73.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 73.0%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 73.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 73.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 73.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 5.2e57 < t

   1. Initial program 16.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 16.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 16.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 11.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 11.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 16.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 16.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 16.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 42.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 42.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 77.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 77.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 77.0%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 76.9%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 76.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 76.9%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 77.0%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 77.0%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 79.2%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 61.3%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
   13. Step-by-step derivation

       1. +-commutative 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

       2. associate--l+ 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

       3. distribute-rgt-out-- 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

       4. metadata-eval 61.3%

          \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

   14. Simplified 61.3%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]
   15. Step-by-step derivation

       1. expm1-log1p-u 60.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}{t}\right)\right)} \]

       2. expm1-udef 60.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}{t}\right)} - 1} \]

   16. Applied egg-rr 56.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(2 \cdot {k}^{-2}\right) \cdot \mathsf{fma}\left({\ell}^{2}, {k}^{-2}, {\ell}^{2} \cdot -0.16666666666666666\right)}{t}\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 58.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(2 \cdot {k}^{-2}\right) \cdot \mathsf{fma}\left({\ell}^{2}, {k}^{-2}, {\ell}^{2} \cdot -0.16666666666666666\right)}{t}\right)\right)} \]

       2. expm1-log1p 59.1%

          \[\leadsto \color{blue}{\frac{\left(2 \cdot {k}^{-2}\right) \cdot \mathsf{fma}\left({\ell}^{2}, {k}^{-2}, {\ell}^{2} \cdot -0.16666666666666666\right)}{t}} \]

       3. associate-/l* 61.3%

          \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-2}}{\frac{t}{\mathsf{fma}\left({\ell}^{2}, {k}^{-2}, {\ell}^{2} \cdot -0.16666666666666666\right)}}} \]

       4. associate-/r/ 63.6%

          \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-2}}{t} \cdot \mathsf{fma}\left({\ell}^{2}, {k}^{-2}, {\ell}^{2} \cdot -0.16666666666666666\right)} \]

       5. fma-udef 63.6%

          \[\leadsto \frac{2 \cdot {k}^{-2}}{t} \cdot \color{blue}{\left({\ell}^{2} \cdot {k}^{-2} + {\ell}^{2} \cdot -0.16666666666666666\right)} \]

       6. distribute-lft-out 70.7%

          \[\leadsto \frac{2 \cdot {k}^{-2}}{t} \cdot \color{blue}{\left({\ell}^{2} \cdot \left({k}^{-2} + -0.16666666666666666\right)\right)} \]

   18. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-2}}{t} \cdot \left({\ell}^{2} \cdot \left({k}^{-2} + -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot k}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {k}^{-2}}{t} \cdot \left({\ell}^{2} \cdot \left({k}^{-2} + -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m \cdot k}{\ell}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2}{t\_m \cdot {k}^{2}}\\ \end{array} \end{array} \]

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 8.4e-166)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
    (if (<= t_m 5.6e+30)
      (/
       (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ (* t_m k) l)))
       (* (/ k t_m) (/ k t_m)))
      (* (/ (pow l 2.0) (pow k 2.0)) (/ 2.0 (* t_m (pow k 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else if (t_m <= 5.6e+30) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) * (2.0 / (t_m * pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.4d-166) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else if (t_m <= 5.6d+30) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
    else
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) * (2.0d0 / (t_m * (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.4e-166) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else if (t_m <= 5.6e+30) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (2.0 / (t_m * Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.4e-166:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	elif t_m <= 5.6e+30:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) * (2.0 / (t_m * math.pow(k, 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.4e-166)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	elseif (t_m <= 5.6e+30)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(Float64(t_m * k) / l))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(2.0 / Float64(t_m * (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.4e-166)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	elseif (t_m <= 5.6e+30)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	else
		tmp = ((l ^ 2.0) / (k ^ 2.0)) * (2.0 / (t_m * (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.4e-166], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+30], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.3999999999999998e-166

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 35.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 35.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 19.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 29.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 35.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 43.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 64.1%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 5.59999999999999966e30

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 48.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 48.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 48.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 48.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 48.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 48.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 48.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 48.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 48.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 48.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 50.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 50.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 50.6%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 63.8%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 70.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 70.6%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 70.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 70.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 70.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 64.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 5.59999999999999966e30 < t

   1. Initial program 26.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 26.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 26.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 22.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 22.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 26.4%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 26.4%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 48.4%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 78.4%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 78.4%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 78.3%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   8. Taylor expanded in k around 0 72.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot k}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{2}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166} \lor \neg \left(t\_m \leq 4.2 \cdot 10^{+79}\right):\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m \cdot k}{\ell}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \end{array} \end{array} \]

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 (<= t_m 8.4e-166) (not (<= t_m 4.2e+79)))
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0))))
    (/
     (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ (* t_m k) l)))
     (* (/ k t_m) (/ k t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 8.4e-166) || !(t_m <= 4.2e+79)) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	} else {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 8.4d-166) .or. (.not. (t_m <= 4.2d+79))) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    else
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 8.4e-166) || !(t_m <= 4.2e+79)) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 8.4e-166) or not (t_m <= 4.2e+79):
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 8.4e-166) || !(t_m <= 4.2e+79))
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(Float64(t_m * k) / l))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 8.4e-166) || ~((t_m <= 4.2e+79)))
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	else
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 8.4e-166], N[Not[LessEqual[t$95$m, 4.2e+79]], $MachinePrecision]], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.3999999999999998e-166 or 4.20000000000000016e79 < t

   1. Initial program 30.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 30.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 30.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 17.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 17.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 30.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 30.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 42.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 63.9%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 8.3999999999999998e-166 < t < 4.20000000000000016e79

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 53.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 57.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 67.7%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 75.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 75.0%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.9%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166} \lor \neg \left(t \leq 4.2 \cdot 10^{+79}\right):\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot k}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166} \lor \neg \left(t\_m \leq 10^{+78}\right):\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\frac{\ell}{\tan k}} \cdot \left({t\_m}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \end{array} \end{array} \]

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 (<= t_m 8.4e-166) (not (<= t_m 1e+78)))
    (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t_m))
    (/
     (/ 2.0 (* (/ t_m (/ l (tan k))) (* (pow t_m 2.0) (/ k l))))
     (* (/ k t_m) (/ k t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 8.4e-166) || !(t_m <= 1e+78)) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t_m);
	} else {
		tmp = (2.0 / ((t_m / (l / tan(k))) * (pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 8.4d-166) .or. (.not. (t_m <= 1d+78))) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t_m)
    else
        tmp = (2.0d0 / ((t_m / (l / tan(k))) * ((t_m ** 2.0d0) * (k / l)))) / ((k / t_m) * (k / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 8.4e-166) || !(t_m <= 1e+78)) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t_m);
	} else {
		tmp = (2.0 / ((t_m / (l / Math.tan(k))) * (Math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 8.4e-166) or not (t_m <= 1e+78):
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t_m)
	else:
		tmp = (2.0 / ((t_m / (l / math.tan(k))) * (math.pow(t_m, 2.0) * (k / l)))) / ((k / t_m) * (k / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 8.4e-166) || !(t_m <= 1e+78))
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t_m));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / Float64(l / tan(k))) * Float64((t_m ^ 2.0) * Float64(k / l)))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 8.4e-166) || ~((t_m <= 1e+78)))
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t_m);
	else
		tmp = (2.0 / ((t_m / (l / tan(k))) * ((t_m ^ 2.0) * (k / l)))) / ((k / t_m) * (k / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 8.4e-166], N[Not[LessEqual[t$95$m, 1e+78]], $MachinePrecision]], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.3999999999999998e-166 or 1.00000000000000001e78 < t

   1. Initial program 30.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 30.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 30.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 17.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 17.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 30.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 30.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 42.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 78.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 78.7%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 78.8%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 78.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 78.7%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 78.7%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 78.7%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 78.8%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 61.7%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   13. Step-by-step derivation

       1. associate-/r* 61.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   14. Simplified 61.3%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   if 8.3999999999999998e-166 < t < 1.00000000000000001e78

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 53.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 57.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 67.7%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 75.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 75.0%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 66.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
   10. Step-by-step derivation

       1. associate-/l* 66.9%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{t}^{2}}}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

       2. associate-/r/ 67.0%

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   11. Simplified 67.0%

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{2}\right)} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166} \lor \neg \left(t \leq 10^{+78}\right):\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{\tan k}} \cdot \left({t}^{2} \cdot \frac{k}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.4 \cdot 10^{-166} \lor \neg \left(t\_m \leq 2.75 \cdot 10^{+79}\right):\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t\_m}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t\_m \cdot k}{\ell}}}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}\\ \end{array} \end{array} \]

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 (<= t_m 8.4e-166) (not (<= t_m 2.75e+79)))
    (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t_m))
    (/
     (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k))) (/ (* t_m k) l)))
     (* (/ k t_m) (/ k t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 8.4e-166) || !(t_m <= 2.75e+79)) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t_m);
	} else {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 8.4d-166) .or. (.not. (t_m <= 2.75d+79))) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t_m)
    else
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 8.4e-166) || !(t_m <= 2.75e+79)) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t_m);
	} else {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 8.4e-166) or not (t_m <= 2.75e+79):
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t_m)
	else:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 8.4e-166) || !(t_m <= 2.75e+79))
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t_m));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k))) * Float64(Float64(t_m * k) / l))) / Float64(Float64(k / t_m) * Float64(k / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 8.4e-166) || ~((t_m <= 2.75e+79)))
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t_m);
	else
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k))) * ((t_m * k) / l))) / ((k / t_m) * (k / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 8.4e-166], N[Not[LessEqual[t$95$m, 2.75e+79]], $MachinePrecision]], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.3999999999999998e-166 or 2.75000000000000003e79 < t

   1. Initial program 30.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 30.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 30.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 17.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 17.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 26.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 30.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 30.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 42.7%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 78.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* 78.7%

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac 78.8%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 78.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

      2. div-inv 78.7%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

      3. pow-flip 78.7%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

      4. metadata-eval 78.7%

         \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

   9. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 78.8%

          \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

   12. Taylor expanded in k around 0 61.7%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   13. Step-by-step derivation

       1. associate-/r* 61.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   14. Simplified 61.3%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   if 8.3999999999999998e-166 < t < 2.75000000000000003e79

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/l* 53.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      9. distribute-frac-neg 53.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      10. unpow2 53.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      11. associate--l+ 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      12. metadata-eval 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      14. unpow2 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      15. distribute-frac-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\frac{-k}{t} \cdot \color{blue}{\left(-\frac{k}{t}\right)}} \]

      16. distribute-frac-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \left(-\frac{k}{t}\right)} \]

      17. sqr-neg 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      18. unpow2 57.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 57.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. times-frac 67.7%

         \[\leadsto \frac{\frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      3. times-frac 75.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

      4. pow2 75.0%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2}}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   9. Taylor expanded in k around 0 67.9%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{k}{t} \cdot \frac{k}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-166} \lor \neg \left(t \leq 2.75 \cdot 10^{+79}\right):\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot k}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 32.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* -0.3333333333333333 (/ (pow l 2.0) (* t_m (pow k 2.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (-0.3333333333333333 * (pow(l, 2.0) / (t_m * pow(k, 2.0))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((-0.3333333333333333d0) * ((l ** 2.0d0) / (t_m * (k ** 2.0d0))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (-0.3333333333333333 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (-0.3333333333333333 * (math.pow(l, 2.0) / (t_m * math.pow(k, 2.0))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(-0.3333333333333333 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0)))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (-0.3333333333333333 * ((l ^ 2.0) / (t_m * (k ^ 2.0))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(-0.3333333333333333 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 35.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 35.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 35.5%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 45.7%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 76.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 76.3%

      \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* 76.3%

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 76.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

7. Simplified 76.3%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
8. Step-by-step derivation

   1. associate-*l/ 76.3%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

   2. div-inv 76.3%

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

   3. pow-flip 76.3%

      \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

   4. metadata-eval 76.3%

      \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

9. Applied egg-rr 76.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
10. Step-by-step derivation

    1. times-frac 76.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

11. Simplified 76.4%

    \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

12. Taylor expanded in k around 0 52.8%

    \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(-0.5 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - -0.3333333333333333 \cdot {\ell}^{2}}}{t} \]
13. Step-by-step derivation

    1. +-commutative 52.8%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} + -0.5 \cdot {\ell}^{2}\right)} - -0.3333333333333333 \cdot {\ell}^{2}}{t} \]

    2. associate--l+ 52.8%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + \left(-0.5 \cdot {\ell}^{2} - -0.3333333333333333 \cdot {\ell}^{2}\right)}}{t} \]

    3. distribute-rgt-out-- 52.8%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + \color{blue}{{\ell}^{2} \cdot \left(-0.5 - -0.3333333333333333\right)}}{t} \]

    4. metadata-eval 52.8%

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot \color{blue}{-0.16666666666666666}}{t} \]

14. Simplified 52.8%

    \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + {\ell}^{2} \cdot -0.16666666666666666}}{t} \]

15. Taylor expanded in k around inf 37.3%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

16. Final simplification 37.3%

    \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}} \]
17. Add Preprocessing

Alternative 24: 60.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot {k}^{-4}\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (* (/ (pow l 2.0) t_m) (pow k -4.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) / t_m) * pow(k, -4.0)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) / t_m) * (k ** (-4.0d0))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / t_m) * Math.pow(k, -4.0)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) / t_m) * math.pow(k, -4.0)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * (k ^ -4.0))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) / t_m) * (k ^ -4.0)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 35.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 35.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 35.5%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 45.7%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 60.2%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. *-commutative 60.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   2. associate-/r* 59.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

7. Simplified 59.8%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
8. Step-by-step derivation

   1. div-inv 59.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

   2. pow-flip 59.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \]

   3. metadata-eval 59.5%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right) \]

9. Applied egg-rr 59.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} \]

10. Final simplification 59.5%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right) \]
11. Add Preprocessing

Alternative 25: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k 4.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0)))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 35.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 35.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 35.5%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 45.7%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 60.2%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. *-commutative 60.2%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   2. associate-/r* 59.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

7. Simplified 59.8%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

8. Final simplification 59.8%

   \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]
9. Add Preprocessing

Alternative 26: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t\_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t_m))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t_m));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t_m))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t_m));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t_m))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t_m)))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t_m));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 35.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 35.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 35.5%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 45.7%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 76.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 76.3%

      \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* 76.3%

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 76.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

7. Simplified 76.3%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]
8. Step-by-step derivation

   1. associate-*l/ 76.3%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}}{{k}^{2} \cdot t}} \]

   2. div-inv 76.3%

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{1}{{\sin k}^{2}}\right)}}{{k}^{2} \cdot t} \]

   3. pow-flip 76.3%

      \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot \color{blue}{{\sin k}^{\left(-2\right)}}\right)}{{k}^{2} \cdot t} \]

   4. metadata-eval 76.3%

      \[\leadsto \frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{\color{blue}{-2}}\right)}{{k}^{2} \cdot t} \]

9. Applied egg-rr 76.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}\right)}{{k}^{2} \cdot t}} \]
10. Step-by-step derivation

    1. times-frac 76.4%

       \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

11. Simplified 76.4%

    \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot {\sin k}^{-2}}{t}} \]

12. Taylor expanded in k around 0 60.2%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
13. Step-by-step derivation

    1. associate-/r* 59.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

14. Simplified 59.9%

    \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

15. Final simplification 59.9%

    \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t} \]
16. Add Preprocessing

Alternative 27: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 35.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 35.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 35.5%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 24.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 32.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 35.5%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 45.7%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 60.2%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

6. Final simplification 60.2%

   \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(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))))