Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.8% → 88.0%

Time: 21.2s

Alternatives: 22

Speedup: 32.4×

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% 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: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t\_m \cdot t\_m}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.45 \cdot 10^{+231}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(t\_2 \cdot \left(\sin k \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)}}{t\_m}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 2.6e+55)
      (/
       2.0
       (/
        (*
         (*
          t_m
          (fma k (* k (/ (tan k) l)) (* (/ (* t_m t_m) l) (* 2.0 (tan k)))))
         (sin k))
        l))
      (if (<= t_m 2.45e+231)
        (/
         2.0
         (*
          (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
          (* (tan k) (* t_2 (* (sin k) t_2)))))
        (/
         (/
          (/
           2.0
           (*
            (+ 2.0 (/ k (/ (* t_m t_m) k)))
            (* (* (tan k) (sin k)) (/ t_m l))))
          t_m)
         (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 2.6e+55) {
		tmp = 2.0 / (((t_m * fma(k, (k * (tan(k) / l)), (((t_m * t_m) / l) * (2.0 * tan(k))))) * sin(k)) / l);
	} else if (t_m <= 2.45e+231) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * (t_2 * (sin(k) * t_2))));
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 2.6e+55)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * fma(k, Float64(k * Float64(tan(k) / l)), Float64(Float64(Float64(t_m * t_m) / l) * Float64(2.0 * tan(k))))) * sin(k)) / l));
	elseif (t_m <= 2.45e+231)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(t_2 * Float64(sin(k) * t_2)))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(Float64(tan(k) * sin(k)) * Float64(t_m / l)))) / t_m) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e+55], N[(2.0 / N[(N[(N[(t$95$m * N[(k * N[(k * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.45e+231], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.6e55

   1. Initial program 61.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 66.6%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \ell\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \sin k\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \sin k\right), \ell\right)\right) \]

   8. Applied egg-rr 54.5%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)}, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({t}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       15. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       17. cos-lowering-cos.f64 85.3%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   11. Simplified 85.3%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{\left(k \cdot k\right) \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right) + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\ell \cdot \cos k}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\cos k \cdot \ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\frac{\sin k}{\cos k}}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. tan-quot N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\tan k}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\tan k, \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k} \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. times-frac N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. tan-quot N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   13. Applied egg-rr 92.1%

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

   if 2.6e55 < t < 2.4500000000000001e231

   1. Initial program 54.5%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left({t}^{\left(\frac{3}{2}\right)}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. metadata-eval 88.9%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 88.9%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \frac{{t}^{\frac{3}{2}}}{\ell}\right) \cdot \sin k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell}\right), \left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({t}^{\frac{3}{2}}\right), \ell\right), \left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \left(\frac{{t}^{\frac{3}{2}}}{\ell} \cdot \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{{t}^{\frac{3}{2}}}{\ell}\right), \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({t}^{\frac{3}{2}}\right), \ell\right), \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \sin k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. sin-lowering-sin.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), \mathsf{sin.f64}\left(k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   6. Applied egg-rr 98.8%

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

   if 2.4500000000000001e231 < t

   1. Initial program 45.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 57.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 58.0%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}}{t}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   10. Applied egg-rr 88.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t \cdot t}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+231}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}{t}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t\_m \cdot t\_m}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left({t\_m}^{0.75} \cdot \frac{{t\_m}^{0.75}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e+121)
    (/
     2.0
     (/
      (*
       (*
        t_m
        (fma k (* k (/ (tan k) l)) (* (/ (* t_m t_m) l) (* 2.0 (tan k)))))
       (sin k))
      l))
    (/
     2.0
     (*
      (* (tan k) (* (sin k) (pow (* (pow t_m 0.75) (/ (pow t_m 0.75) l)) 2.0)))
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 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 <= 5e+121) {
		tmp = 2.0 / (((t_m * fma(k, (k * (tan(k) / l)), (((t_m * t_m) / l) * (2.0 * tan(k))))) * sin(k)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 0.75) * (pow(t_m, 0.75) / l)), 2.0))) * (1.0 + (1.0 + pow((k / t_m), 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 <= 5e+121)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * fma(k, Float64(k * Float64(tan(k) / l)), Float64(Float64(Float64(t_m * t_m) / l) * Float64(2.0 * tan(k))))) * sin(k)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 0.75) * Float64((t_m ^ 0.75) / l)) ^ 2.0))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e+121], N[(2.0 / N[(N[(N[(t$95$m * N[(k * N[(k * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[(N[Power[t$95$m, 0.75], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000007e121

   1. Initial program 61.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 59.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 66.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \ell\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \sin k\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \sin k\right), \ell\right)\right) \]

   8. Applied egg-rr 56.1%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)}, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({t}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       15. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       17. cos-lowering-cos.f64 86.2%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   11. Simplified 86.2%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{\left(k \cdot k\right) \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right) + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\ell \cdot \cos k}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\cos k \cdot \ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\frac{\sin k}{\cos k}}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. tan-quot N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\tan k}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\tan k, \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k} \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. times-frac N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. tan-quot N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   13. Applied egg-rr 92.6%

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

   if 5.00000000000000007e121 < t

   1. Initial program 44.9%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left({t}^{\left(\frac{3}{2}\right)}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. metadata-eval 77.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 77.2%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)}\right), \left(\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{\frac{3}{2}}{2}\right)\right), \left(\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{4}\right), \left(\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{4}\right), \mathsf{/.f64}\left(\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)}\right), \ell\right)\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{4}\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{\frac{3}{2}}{2}\right)\right), \ell\right)\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. metadata-eval 87.6%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{4}\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{4}\right), \ell\right)\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   6. Applied egg-rr 87.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t \cdot t}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t\_m \cdot t\_m}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+231}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)}}{t\_m}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e+121)
    (/
     2.0
     (/
      (*
       (*
        t_m
        (fma k (* k (/ (tan k) l)) (* (/ (* t_m t_m) l) (* 2.0 (tan k)))))
       (sin k))
      l))
    (if (<= t_m 5.1e+231)
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))))
      (/
       (/
        (/
         2.0
         (* (+ 2.0 (/ k (/ (* t_m t_m) k))) (* (* (tan k) (sin k)) (/ t_m l))))
        t_m)
       (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e+121) {
		tmp = 2.0 / (((t_m * fma(k, (k * (tan(k) / l)), (((t_m * t_m) / l) * (2.0 * tan(k))))) * sin(k)) / l);
	} else if (t_m <= 5.1e+231) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))));
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e+121)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * fma(k, Float64(k * Float64(tan(k) / l)), Float64(Float64(Float64(t_m * t_m) / l) * Float64(2.0 * tan(k))))) * sin(k)) / l));
	elseif (t_m <= 5.1e+231)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(Float64(tan(k) * sin(k)) * Float64(t_m / l)))) / t_m) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e+121], N[(2.0 / N[(N[(N[(t$95$m * N[(k * N[(k * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.1e+231], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.49999999999999965e121

   1. Initial program 61.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 59.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 66.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \ell\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \sin k\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \sin k\right), \ell\right)\right) \]

   8. Applied egg-rr 56.1%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)}, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({t}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       15. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       17. cos-lowering-cos.f64 86.2%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   11. Simplified 86.2%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{\left(k \cdot k\right) \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right) + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\ell \cdot \cos k}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\cos k \cdot \ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\frac{\sin k}{\cos k}}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. tan-quot N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\tan k}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\tan k, \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k} \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. times-frac N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. tan-quot N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   13. Applied egg-rr 92.6%

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

   if 7.49999999999999965e121 < t < 5.0999999999999997e231

   1. Initial program 44.8%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left({t}^{\left(\frac{3}{2}\right)}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. metadata-eval 90.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.7%

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

   if 5.0999999999999997e231 < t

   1. Initial program 45.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 57.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 58.0%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}}{t}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   10. Applied egg-rr 88.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t \cdot t}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+231}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}{t}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(\sin k \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + t\_2\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+230}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)}}{t\_m}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ k t_m) (/ t_m k))))
   (*
    t_s
    (if (<= t_m 3.5e-100)
      (/ 2.0 (/ (* (sin k) (* (sin k) (/ (* t_m (/ (* k k) l)) l))) (cos k)))
      (if (<= t_m 6.8e+90)
        (/
         (/ (/ 2.0 (* t_m (* t_m t_m))) (/ (sin k) l))
         (/ (* (tan k) (+ 2.0 t_2)) l))
        (if (<= t_m 4.8e+230)
          (/
           2.0
           (*
            (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
            (+ 1.0 (+ 1.0 t_2))))
          (/
           (/
            (/
             2.0
             (*
              (+ 2.0 (/ k (/ (* t_m t_m) k)))
              (* (* (tan k) (sin k)) (/ t_m l))))
            t_m)
           (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k / t_m) / (t_m / k);
	double tmp;
	if (t_m <= 3.5e-100) {
		tmp = 2.0 / ((sin(k) * (sin(k) * ((t_m * ((k * k) / l)) / l))) / cos(k));
	} else if (t_m <= 6.8e+90) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + t_2)) / l);
	} else if (t_m <= 4.8e+230) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + t_2)));
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) / (t_m / k)
    if (t_m <= 3.5d-100) then
        tmp = 2.0d0 / ((sin(k) * (sin(k) * ((t_m * ((k * k) / l)) / l))) / cos(k))
    else if (t_m <= 6.8d+90) then
        tmp = ((2.0d0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0d0 + t_2)) / l)
    else if (t_m <= 4.8d+230) then
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0))) * (1.0d0 + (1.0d0 + t_2)))
    else
        tmp = ((2.0d0 / ((2.0d0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k / t_m) / (t_m / k);
	double tmp;
	if (t_m <= 3.5e-100) {
		tmp = 2.0 / ((Math.sin(k) * (Math.sin(k) * ((t_m * ((k * k) / l)) / l))) / Math.cos(k));
	} else if (t_m <= 6.8e+90) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (Math.sin(k) / l)) / ((Math.tan(k) * (2.0 + t_2)) / l);
	} else if (t_m <= 4.8e+230) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + t_2)));
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((Math.tan(k) * Math.sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k / t_m) / (t_m / k)
	tmp = 0
	if t_m <= 3.5e-100:
		tmp = 2.0 / ((math.sin(k) * (math.sin(k) * ((t_m * ((k * k) / l)) / l))) / math.cos(k))
	elif t_m <= 6.8e+90:
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (math.sin(k) / l)) / ((math.tan(k) * (2.0 + t_2)) / l)
	elif t_m <= 4.8e+230:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + t_2)))
	else:
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((math.tan(k) * math.sin(k)) * (t_m / l)))) / t_m) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k / t_m) / Float64(t_m / k))
	tmp = 0.0
	if (t_m <= 3.5e-100)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(sin(k) * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l))) / cos(k)));
	elseif (t_m <= 6.8e+90)
		tmp = Float64(Float64(Float64(2.0 / Float64(t_m * Float64(t_m * t_m))) / Float64(sin(k) / l)) / Float64(Float64(tan(k) * Float64(2.0 + t_2)) / l));
	elseif (t_m <= 4.8e+230)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(1.0 + t_2))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(Float64(tan(k) * sin(k)) * Float64(t_m / l)))) / t_m) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) / (t_m / k);
	tmp = 0.0;
	if (t_m <= 3.5e-100)
		tmp = 2.0 / ((sin(k) * (sin(k) * ((t_m * ((k * k) / l)) / l))) / cos(k));
	elseif (t_m <= 6.8e+90)
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + t_2)) / l);
	elseif (t_m <= 4.8e+230)
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))) * (1.0 + (1.0 + t_2)));
	else
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-100], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e+90], N[(N[(N[(2.0 / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+230], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.5000000000000001e-100

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right), \mathsf{cos.f64}\left(\color{blue}{k}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\sin k \cdot \sin k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\sin k \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right), \mathsf{cos.f64}\left(\color{blue}{k}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin k, \left(\sin k \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right), \mathsf{cos.f64}\left(\color{blue}{k}\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\sin k \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\sin k, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\ell}\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      14. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   9. Applied egg-rr 72.6%

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

   if 3.5000000000000001e-100 < t < 6.80000000000000036e90

   1. Initial program 83.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 83.3%

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

      1. associate-/r* N/A

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

      2. times-frac N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t \cdot \left(t \cdot t\right)}\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k} \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\sin k, \ell\right)\right), \left(\frac{\tan k \cdot \color{blue}{\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \left(\frac{\tan k \cdot \left(\color{blue}{2} + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 94.1%

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

   if 6.80000000000000036e90 < t < 4.79999999999999996e230

   1. Initial program 45.9%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left({t}^{\left(\frac{3}{2}\right)}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. metadata-eval 89.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 89.2%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right), 1\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{k}{t} \cdot \frac{1}{\frac{t}{k}}\right)\right), 1\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), 1\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{k}{t}\right), \left(\frac{t}{k}\right)\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), \left(\frac{t}{k}\right)\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 89.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), \mathsf{/.f64}\left(t, k\right)\right)\right), 1\right)\right)\right) \]

   6. Applied egg-rr 89.2%

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

   if 4.79999999999999996e230 < t

   1. Initial program 45.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 57.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 58.0%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}}{t}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   10. Applied egg-rr 88.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(\sin k \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+230}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}{t}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t\_m \cdot t\_m}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{+230}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)}}{t\_m}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.5e+121)
    (/
     2.0
     (/
      (*
       (*
        t_m
        (fma k (* k (/ (tan k) l)) (* (/ (* t_m t_m) l) (* 2.0 (tan k)))))
       (sin k))
      l))
    (if (<= t_m 6.8e+230)
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
        (+ 1.0 (+ 1.0 (/ (/ k t_m) (/ t_m k))))))
      (/
       (/
        (/
         2.0
         (* (+ 2.0 (/ k (/ (* t_m t_m) k))) (* (* (tan k) (sin k)) (/ t_m l))))
        t_m)
       (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.5e+121) {
		tmp = 2.0 / (((t_m * fma(k, (k * (tan(k) / l)), (((t_m * t_m) / l) * (2.0 * tan(k))))) * sin(k)) / l);
	} else if (t_m <= 6.8e+230) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (1.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.5e+121)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * fma(k, Float64(k * Float64(tan(k) / l)), Float64(Float64(Float64(t_m * t_m) / l) * Float64(2.0 * tan(k))))) * sin(k)) / l));
	elseif (t_m <= 6.8e+230)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(Float64(tan(k) * sin(k)) * Float64(t_m / l)))) / t_m) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.5e+121], N[(2.0 / N[(N[(N[(t$95$m * N[(k * N[(k * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e+230], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.49999999999999949e121

   1. Initial program 61.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 59.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 66.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \ell\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \sin k\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \sin k\right), \ell\right)\right) \]

   8. Applied egg-rr 56.1%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)}, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({t}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       15. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       17. cos-lowering-cos.f64 86.2%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   11. Simplified 86.2%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{\left(k \cdot k\right) \cdot \sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \cos k} + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right) + 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. fma-define N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\mathsf{fma}\left(k, k \cdot \frac{\sin k}{\ell \cdot \cos k}, 2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \left(k \cdot \frac{\sin k}{\ell \cdot \cos k}\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\ell \cdot \cos k}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\sin k}{\cos k \cdot \ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\frac{\sin k}{\cos k}}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. tan-quot N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\tan k}{\ell}\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\tan k, \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(2 \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell \cdot \cos k} \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. times-frac N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. tan-quot N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{fma.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right)\right), \left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   13. Applied egg-rr 92.6%

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

   if 9.49999999999999949e121 < t < 6.79999999999999973e230

   1. Initial program 44.8%

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

      1. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left({t}^{\left(\frac{3}{2}\right)}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. metadata-eval 90.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.7%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right), 1\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{k}{t} \cdot \frac{1}{\frac{t}{k}}\right)\right), 1\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), 1\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{k}{t}\right), \left(\frac{t}{k}\right)\right)\right), 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), \left(\frac{t}{k}\right)\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 90.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(t, \frac{3}{2}\right), \ell\right), 2\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), \mathsf{/.f64}\left(t, k\right)\right)\right), 1\right)\right)\right) \]

   6. Applied egg-rr 90.7%

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

   if 6.79999999999999973e230 < t

   1. Initial program 45.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 57.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 58.0%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}}{t}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   10. Applied egg-rr 88.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \mathsf{fma}\left(k, k \cdot \frac{\tan k}{\ell}, \frac{t \cdot t}{\ell} \cdot \left(2 \cdot \tan k\right)\right)\right) \cdot \sin k}{\ell}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+230}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}{t}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(\sin k \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)}}{t\_m}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.3e-101)
    (/ 2.0 (/ (* (sin k) (* (sin k) (/ (* t_m (/ (* k k) l)) l))) (cos k)))
    (if (<= t_m 5.6e+102)
      (/
       (/ (/ 2.0 (* t_m (* t_m t_m))) (/ (sin k) l))
       (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l))
      (/
       (/
        (/
         2.0
         (* (+ 2.0 (/ k (/ (* t_m t_m) k))) (* (* (tan k) (sin k)) (/ t_m l))))
        t_m)
       (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.3e-101) {
		tmp = 2.0 / ((sin(k) * (sin(k) * ((t_m * ((k * k) / l)) / l))) / cos(k));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.3d-101) then
        tmp = 2.0d0 / ((sin(k) * (sin(k) * ((t_m * ((k * k) / l)) / l))) / cos(k))
    else if (t_m <= 5.6d+102) then
        tmp = ((2.0d0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l)
    else
        tmp = ((2.0d0 / ((2.0d0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.3e-101) {
		tmp = 2.0 / ((Math.sin(k) * (Math.sin(k) * ((t_m * ((k * k) / l)) / l))) / Math.cos(k));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (Math.sin(k) / l)) / ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((Math.tan(k) * Math.sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.3e-101:
		tmp = 2.0 / ((math.sin(k) * (math.sin(k) * ((t_m * ((k * k) / l)) / l))) / math.cos(k))
	elif t_m <= 5.6e+102:
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (math.sin(k) / l)) / ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)
	else:
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((math.tan(k) * math.sin(k)) * (t_m / l)))) / t_m) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.3e-101)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(sin(k) * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l))) / cos(k)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(t_m * Float64(t_m * t_m))) / Float64(sin(k) / l)) / Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(Float64(tan(k) * sin(k)) * Float64(t_m / l)))) / t_m) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.3e-101)
		tmp = 2.0 / ((sin(k) * (sin(k) * ((t_m * ((k * k) / l)) / l))) / cos(k));
	elseif (t_m <= 5.6e+102)
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	else
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * ((tan(k) * sin(k)) * (t_m / l)))) / t_m) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.3e-101], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.3000000000000006e-101

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right), \mathsf{cos.f64}\left(\color{blue}{k}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\sin k \cdot \sin k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\sin k \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right), \mathsf{cos.f64}\left(\color{blue}{k}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin k, \left(\sin k \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right), \mathsf{cos.f64}\left(\color{blue}{k}\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\sin k \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\sin k, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\ell}\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      14. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   9. Applied egg-rr 72.6%

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

   if 8.3000000000000006e-101 < t < 5.60000000000000037e102

   1. Initial program 84.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 81.8%

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

      1. associate-/r* N/A

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

      2. times-frac N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t \cdot \left(t \cdot t\right)}\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k} \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\sin k, \ell\right)\right), \left(\frac{\tan k \cdot \color{blue}{\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \left(\frac{\tan k \cdot \left(\color{blue}{2} + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 94.4%

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

   if 5.60000000000000037e102 < t

   1. Initial program 43.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 48.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 60.7%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}}{t}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   10. Applied egg-rr 78.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(\sin k \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}{t}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(t\_2 \cdot \frac{t\_m}{\ell}\right)}}{t\_m}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (sin k))))
   (*
    t_s
    (if (<= t_m 3.5e-100)
      (/ 2.0 (* t_2 (/ (* t_m (/ (* k k) l)) l)))
      (if (<= t_m 5.6e+102)
        (/
         (/ (/ 2.0 (* t_m (* t_m t_m))) (/ (sin k) l))
         (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l))
        (/
         (/ (/ 2.0 (* (+ 2.0 (/ k (/ (* t_m t_m) k))) (* t_2 (/ t_m l)))) t_m)
         (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * sin(k);
	double tmp;
	if (t_m <= 3.5e-100) {
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = tan(k) * sin(k)
    if (t_m <= 3.5d-100) then
        tmp = 2.0d0 / (t_2 * ((t_m * ((k * k) / l)) / l))
    else if (t_m <= 5.6d+102) then
        tmp = ((2.0d0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l)
    else
        tmp = ((2.0d0 / ((2.0d0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) / t_m) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * Math.sin(k);
	double tmp;
	if (t_m <= 3.5e-100) {
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (Math.sin(k) / l)) / ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	} else {
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) / t_m) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.tan(k) * math.sin(k)
	tmp = 0
	if t_m <= 3.5e-100:
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l))
	elif t_m <= 5.6e+102:
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (math.sin(k) / l)) / ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)
	else:
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) / t_m) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * sin(k))
	tmp = 0.0
	if (t_m <= 3.5e-100)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(t_m * Float64(t_m * t_m))) / Float64(sin(k) / l)) / Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(t_2 * Float64(t_m / l)))) / t_m) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = tan(k) * sin(k);
	tmp = 0.0;
	if (t_m <= 3.5e-100)
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	elseif (t_m <= 5.6e+102)
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	else
		tmp = ((2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) / t_m) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-100], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.5000000000000001e-100

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 72.6%

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

   if 3.5000000000000001e-100 < t < 5.60000000000000037e102

   1. Initial program 84.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 81.8%

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

      1. associate-/r* N/A

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

      2. times-frac N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t \cdot \left(t \cdot t\right)}\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k} \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\sin k, \ell\right)\right), \left(\frac{\tan k \cdot \color{blue}{\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \left(\frac{\tan k \cdot \left(\color{blue}{2} + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 94.4%

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

   if 5.60000000000000037e102 < t

   1. Initial program 43.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 48.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 60.7%

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

      1. associate-/r* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}}{t}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   10. Applied egg-rr 78.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}{t}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 6.9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot t\_2\right)}{\ell} \cdot \left(t\_m \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (sin k))))
   (*
    t_s
    (if (<= t_m 2.9e-100)
      (/ 2.0 (* t_2 (/ (* t_m (/ (* k k) l)) l)))
      (if (<= t_m 6.9e+102)
        (/
         (/ (/ 2.0 (* t_m (* t_m t_m))) (/ (sin k) l))
         (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l))
        (/
         2.0
         (*
          (/ (* t_m (* (+ 2.0 (/ k (/ (* t_m t_m) k))) t_2)) l)
          (* t_m (/ t_m l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * sin(k);
	double tmp;
	if (t_m <= 2.9e-100) {
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 6.9e+102) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	} else {
		tmp = 2.0 / (((t_m * ((2.0 + (k / ((t_m * t_m) / k))) * t_2)) / l) * (t_m * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = tan(k) * sin(k)
    if (t_m <= 2.9d-100) then
        tmp = 2.0d0 / (t_2 * ((t_m * ((k * k) / l)) / l))
    else if (t_m <= 6.9d+102) then
        tmp = ((2.0d0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l)
    else
        tmp = 2.0d0 / (((t_m * ((2.0d0 + (k / ((t_m * t_m) / k))) * t_2)) / l) * (t_m * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * Math.sin(k);
	double tmp;
	if (t_m <= 2.9e-100) {
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 6.9e+102) {
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (Math.sin(k) / l)) / ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	} else {
		tmp = 2.0 / (((t_m * ((2.0 + (k / ((t_m * t_m) / k))) * t_2)) / l) * (t_m * (t_m / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.tan(k) * math.sin(k)
	tmp = 0
	if t_m <= 2.9e-100:
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l))
	elif t_m <= 6.9e+102:
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (math.sin(k) / l)) / ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)
	else:
		tmp = 2.0 / (((t_m * ((2.0 + (k / ((t_m * t_m) / k))) * t_2)) / l) * (t_m * (t_m / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * sin(k))
	tmp = 0.0
	if (t_m <= 2.9e-100)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	elseif (t_m <= 6.9e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(t_m * Float64(t_m * t_m))) / Float64(sin(k) / l)) / Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * t_2)) / l) * Float64(t_m * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = tan(k) * sin(k);
	tmp = 0.0;
	if (t_m <= 2.9e-100)
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	elseif (t_m <= 6.9e+102)
		tmp = ((2.0 / (t_m * (t_m * t_m))) / (sin(k) / l)) / ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l);
	else
		tmp = 2.0 / (((t_m * ((2.0 + (k / ((t_m * t_m) / k))) * t_2)) / l) * (t_m * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-100], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.9e+102], N[(N[(N[(2.0 / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.89999999999999975e-100

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 72.6%

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

   if 2.89999999999999975e-100 < t < 6.89999999999999966e102

   1. Initial program 84.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 81.8%

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

      1. associate-/r* N/A

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

      2. times-frac N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t \cdot \left(t \cdot t\right)}\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\color{blue}{\tan k} \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\sin k}{\ell}\right)\right), \left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\sin k, \ell\right)\right), \left(\frac{\tan k \cdot \color{blue}{\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \left(\frac{\tan k \cdot \left(\color{blue}{2} + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 94.4%

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

   if 6.89999999999999966e102 < t

   1. Initial program 43.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 48.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 60.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \left(\frac{t}{\ell} \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{t}\right)\right)\right) \]

      4. /-lowering-/.f64 78.8%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right)\right)\right) \]

   10. Applied egg-rr 78.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\\ t_3 := \tan k \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{\left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(\tan k \cdot t\_2\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(t\_2 \cdot t\_3\right)}{\ell} \cdot \left(t\_m \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (/ k (/ (* t_m t_m) k)))) (t_3 (* (tan k) (sin k))))
   (*
    t_s
    (if (<= t_m 7.4e-53)
      (/ 2.0 (* t_3 (/ (* t_m (/ (* k k) l)) l)))
      (if (<= t_m 1.22e+101)
        (/ 2.0 (/ (* (sin k) (/ (* (* t_m (* t_m t_m)) (* (tan k) t_2)) l)) l))
        (/ 2.0 (* (/ (* t_m (* t_2 t_3)) l) (* t_m (/ t_m l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + (k / ((t_m * t_m) / k));
	double t_3 = tan(k) * sin(k);
	double tmp;
	if (t_m <= 7.4e-53) {
		tmp = 2.0 / (t_3 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 1.22e+101) {
		tmp = 2.0 / ((sin(k) * (((t_m * (t_m * t_m)) * (tan(k) * t_2)) / l)) / l);
	} else {
		tmp = 2.0 / (((t_m * (t_2 * t_3)) / l) * (t_m * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 + (k / ((t_m * t_m) / k))
    t_3 = tan(k) * sin(k)
    if (t_m <= 7.4d-53) then
        tmp = 2.0d0 / (t_3 * ((t_m * ((k * k) / l)) / l))
    else if (t_m <= 1.22d+101) then
        tmp = 2.0d0 / ((sin(k) * (((t_m * (t_m * t_m)) * (tan(k) * t_2)) / l)) / l)
    else
        tmp = 2.0d0 / (((t_m * (t_2 * t_3)) / l) * (t_m * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + (k / ((t_m * t_m) / k));
	double t_3 = Math.tan(k) * Math.sin(k);
	double tmp;
	if (t_m <= 7.4e-53) {
		tmp = 2.0 / (t_3 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 1.22e+101) {
		tmp = 2.0 / ((Math.sin(k) * (((t_m * (t_m * t_m)) * (Math.tan(k) * t_2)) / l)) / l);
	} else {
		tmp = 2.0 / (((t_m * (t_2 * t_3)) / l) * (t_m * (t_m / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 + (k / ((t_m * t_m) / k))
	t_3 = math.tan(k) * math.sin(k)
	tmp = 0
	if t_m <= 7.4e-53:
		tmp = 2.0 / (t_3 * ((t_m * ((k * k) / l)) / l))
	elif t_m <= 1.22e+101:
		tmp = 2.0 / ((math.sin(k) * (((t_m * (t_m * t_m)) * (math.tan(k) * t_2)) / l)) / l)
	else:
		tmp = 2.0 / (((t_m * (t_2 * t_3)) / l) * (t_m * (t_m / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))
	t_3 = Float64(tan(k) * sin(k))
	tmp = 0.0
	if (t_m <= 7.4e-53)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	elseif (t_m <= 1.22e+101)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(Float64(t_m * Float64(t_m * t_m)) * Float64(tan(k) * t_2)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(t_2 * t_3)) / l) * Float64(t_m * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 + (k / ((t_m * t_m) / k));
	t_3 = tan(k) * sin(k);
	tmp = 0.0;
	if (t_m <= 7.4e-53)
		tmp = 2.0 / (t_3 * ((t_m * ((k * k) / l)) / l));
	elseif (t_m <= 1.22e+101)
		tmp = 2.0 / ((sin(k) * (((t_m * (t_m * t_m)) * (tan(k) * t_2)) / l)) / l);
	else
		tmp = 2.0 / (((t_m * (t_2 * t_3)) / l) * (t_m * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.4e-53], N[(2.0 / N[(t$95$3 * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.22e+101], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.39999999999999965e-53

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.8%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 73.1%

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

   if 7.39999999999999965e-53 < t < 1.22e101

   1. Initial program 90.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 87.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 93.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \ell\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \sin k\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \sin k\right), \ell\right)\right) \]

   8. Applied egg-rr 94.7%

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

   if 1.22e101 < t

   1. Initial program 43.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 38.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 48.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 60.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \left(\frac{t}{\ell} \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{t}\right)\right)\right) \]

      4. /-lowering-/.f64 78.8%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), t\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right)\right)\right) \]

   10. Applied egg-rr 78.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\tan k \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-54)
    (/ 2.0 (* (* (tan k) (sin k)) (/ (* t_m (/ (* k k) l)) l)))
    (if (<= t_m 3.9e+99)
      (/
       2.0
       (/
        (*
         (* t_m (* t_m t_m))
         (* (sin k) (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l)))
        l))
      (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-54) {
		tmp = 2.0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 3.9e+99) {
		tmp = 2.0 / (((t_m * (t_m * t_m)) * (sin(k) * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l))) / l);
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.1d-54) then
        tmp = 2.0d0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l))
    else if (t_m <= 3.9d+99) then
        tmp = 2.0d0 / (((t_m * (t_m * t_m)) * (sin(k) * ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l))) / l)
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-54) {
		tmp = 2.0 / ((Math.tan(k) * Math.sin(k)) * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 3.9e+99) {
		tmp = 2.0 / (((t_m * (t_m * t_m)) * (Math.sin(k) * ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l))) / l);
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.1e-54:
		tmp = 2.0 / ((math.tan(k) * math.sin(k)) * ((t_m * ((k * k) / l)) / l))
	elif t_m <= 3.9e+99:
		tmp = 2.0 / (((t_m * (t_m * t_m)) * (math.sin(k) * ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l))) / l)
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.1e-54)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * sin(k)) * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	elseif (t_m <= 3.9e+99)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(t_m * t_m)) * Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l))) / l));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.1e-54)
		tmp = 2.0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l));
	elseif (t_m <= 3.9e+99)
		tmp = 2.0 / (((t_m * (t_m * t_m)) * (sin(k) * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l))) / l);
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-54], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.9e+99], N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.10000000000000004e-54

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.8%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 73.1%

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

   if 3.10000000000000004e-54 < t < 3.89999999999999995e99

   1. Initial program 90.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 89.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 96.6%

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

   if 3.89999999999999995e99 < t

   1. Initial program 44.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 37.9%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 44.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 44.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 51.9%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 60.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 76.4%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-39)
    (/ 2.0 (* (* (tan k) (sin k)) (/ (* t_m (/ (* k k) l)) l)))
    (if (<= t_m 2e+99)
      (*
       l
       (/
        (/ 2.0 (* t_m (* t_m t_m)))
        (* (sin k) (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l))))
      (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = 2.0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 2e+99) {
		tmp = l * ((2.0 / (t_m * (t_m * t_m))) / (sin(k) * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.75d-39) then
        tmp = 2.0d0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l))
    else if (t_m <= 2d+99) then
        tmp = l * ((2.0d0 / (t_m * (t_m * t_m))) / (sin(k) * ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l)))
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = 2.0 / ((Math.tan(k) * Math.sin(k)) * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 2e+99) {
		tmp = l * ((2.0 / (t_m * (t_m * t_m))) / (Math.sin(k) * ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.75e-39:
		tmp = 2.0 / ((math.tan(k) * math.sin(k)) * ((t_m * ((k * k) / l)) / l))
	elif t_m <= 2e+99:
		tmp = l * ((2.0 / (t_m * (t_m * t_m))) / (math.sin(k) * ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)))
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-39)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * sin(k)) * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	elseif (t_m <= 2e+99)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(t_m * Float64(t_m * t_m))) / Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l))));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.75e-39)
		tmp = 2.0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l));
	elseif (t_m <= 2e+99)
		tmp = l * ((2.0 / (t_m * (t_m * t_m))) / (sin(k) * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)));
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-39], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+99], N[(l * N[(N[(2.0 / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.75000000000000009e-39

   1. Initial program 58.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 56.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 63.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 63.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 73.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 73.9%

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

   if 2.75000000000000009e-39 < t < 1.9999999999999999e99

   1. Initial program 88.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 87.8%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}\right), \color{blue}{\ell}\right) \]

   6. Applied egg-rr 96.1%

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

   if 1.9999999999999999e99 < t

   1. Initial program 44.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 37.9%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 44.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 44.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 51.9%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 60.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 76.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 76.4%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right) \cdot \left(t\_2 \cdot \frac{t\_m}{\ell}\right)} \cdot \frac{\ell}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (sin k))))
   (*
    t_s
    (if (<= t_m 2.7e-99)
      (/ 2.0 (* t_2 (/ (* t_m (/ (* k k) l)) l)))
      (if (<= t_m 8.5e+134)
        (*
         (/ 2.0 (* (+ 2.0 (/ k (/ (* t_m t_m) k))) (* t_2 (/ t_m l))))
         (/ l (* t_m t_m)))
        (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * sin(k);
	double tmp;
	if (t_m <= 2.7e-99) {
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 8.5e+134) {
		tmp = (2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) * (l / (t_m * t_m));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = tan(k) * sin(k)
    if (t_m <= 2.7d-99) then
        tmp = 2.0d0 / (t_2 * ((t_m * ((k * k) / l)) / l))
    else if (t_m <= 8.5d+134) then
        tmp = (2.0d0 / ((2.0d0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) * (l / (t_m * t_m))
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * Math.sin(k);
	double tmp;
	if (t_m <= 2.7e-99) {
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	} else if (t_m <= 8.5e+134) {
		tmp = (2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) * (l / (t_m * t_m));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.tan(k) * math.sin(k)
	tmp = 0
	if t_m <= 2.7e-99:
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l))
	elif t_m <= 8.5e+134:
		tmp = (2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) * (l / (t_m * t_m))
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * sin(k))
	tmp = 0.0
	if (t_m <= 2.7e-99)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	elseif (t_m <= 8.5e+134)
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) * Float64(t_2 * Float64(t_m / l)))) * Float64(l / Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = tan(k) * sin(k);
	tmp = 0.0;
	if (t_m <= 2.7e-99)
		tmp = 2.0 / (t_2 * ((t_m * ((k * k) / l)) / l));
	elseif (t_m <= 8.5e+134)
		tmp = (2.0 / ((2.0 + (k / ((t_m * t_m) / k))) * (t_2 * (t_m / l)))) * (l / (t_m * t_m));
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-99], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+134], N[(N[(2.0 / N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.7e-99

   1. Initial program 57.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 62.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 72.6%

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

   if 2.7e-99 < t < 8.50000000000000024e134

   1. Initial program 76.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 74.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 81.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right) \cdot \color{blue}{\frac{t \cdot t}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\sin k \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot t\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

   8. Applied egg-rr 83.8%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{\left(\frac{2}{\frac{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)\right) \cdot t}{\ell}}\right)}\right) \]

   10. Applied egg-rr 84.5%

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

   if 8.50000000000000024e134 < t

   1. Initial program 45.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 39.9%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 47.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 47.4%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 55.9%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 80.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 80.1%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\ell}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-39)
    (/ 2.0 (* (* (tan k) (sin k)) (/ (* t_m (/ (* k k) l)) l)))
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-39) {
		tmp = 2.0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.8d-39) then
        tmp = 2.0d0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l))
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-39) {
		tmp = 2.0 / ((Math.tan(k) * Math.sin(k)) * ((t_m * ((k * k) / l)) / l));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.8e-39:
		tmp = 2.0 / ((math.tan(k) * math.sin(k)) * ((t_m * ((k * k) / l)) / l))
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-39)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * sin(k)) * Float64(Float64(t_m * Float64(Float64(k * k) / l)) / l)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.8e-39)
		tmp = 2.0 / ((tan(k) * sin(k)) * ((t_m * ((k * k) / l)) / l));
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-39], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-39

   1. Initial program 58.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 56.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 63.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 63.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k}\right)}\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right) \]

      6. tan-quot N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\sin k \cdot \tan k\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \color{blue}{t}}{\ell \cdot \ell}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right), \ell\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \frac{k \cdot k}{\ell}\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell}\right)\right), \ell\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \ell\right)\right)\right) \]

      17. *-lowering-*.f64 73.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \ell\right)\right)\right) \]

   9. Applied egg-rr 73.9%

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

   if 2.8000000000000001e-39 < t

   1. Initial program 59.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.1%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 79.1%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{\frac{k \cdot k}{\frac{\ell}{t\_m}}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-39)
    (/ 2.0 (* (/ (* k k) l) (/ (/ (* k k) (/ l t_m)) (cos k))))
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = 2.0 / (((k * k) / l) * (((k * k) / (l / t_m)) / cos(k)));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.75d-39) then
        tmp = 2.0d0 / (((k * k) / l) * (((k * k) / (l / t_m)) / cos(k)))
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = 2.0 / (((k * k) / l) * (((k * k) / (l / t_m)) / Math.cos(k)));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.75e-39:
		tmp = 2.0 / (((k * k) / l) * (((k * k) / (l / t_m)) / math.cos(k)))
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(Float64(k * k) / Float64(l / t_m)) / cos(k))));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.75e-39)
		tmp = 2.0 / (((k * k) / l) * (((k * k) / (l / t_m)) / cos(k)));
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-39], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.75000000000000009e-39

   1. Initial program 58.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 56.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\sin k}^{2}\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({k}^{2} \cdot t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      14. cos-lowering-cos.f64 63.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 63.7%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left({k}^{2}\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

      2. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   10. Simplified 61.4%

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

       1. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}}{\cos \color{blue}{k}}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{k \cdot k}{\ell} \cdot \frac{1}{\frac{\ell}{\left(k \cdot k\right) \cdot t}}}{\cos k}\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{k \cdot k}{\ell} \cdot \frac{1}{\frac{\frac{\ell}{t}}{k \cdot k}}}{\cos k}\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}{\cos k}\right)\right) \]

       5. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot \color{blue}{\frac{\frac{k \cdot k}{\frac{\ell}{t}}}{\cos k}}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k \cdot k}{\ell}\right), \color{blue}{\left(\frac{\frac{k \cdot k}{\frac{\ell}{t}}}{\cos k}\right)}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right), \left(\frac{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}}}{\cos k}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \left(\frac{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}}}{\cos k}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{k \cdot k}{\frac{\ell}{t}}\right), \color{blue}{\cos k}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(k \cdot k\right), \left(\frac{\ell}{t}\right)\right), \cos \color{blue}{k}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\ell}{t}\right)\right), \cos k\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \cos k\right)\right)\right) \]

       13. cos-lowering-cos.f64 67.7%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right)\right) \]

   12. Applied egg-rr 67.7%

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

   if 2.75000000000000009e-39 < t

   1. Initial program 59.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.1%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 79.1%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.06 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell} + \left(k \cdot k\right) \cdot \left(t\_m \cdot \left(\frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell} \cdot 0.3333333333333333 + \left(\frac{1}{\ell} + \frac{\left(t\_m \cdot t\_m\right) \cdot -0.3333333333333333}{\ell}\right)\right) + \left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot 0.016666666666666666}{\ell} + \frac{0.3333333333333333}{\ell}\right) + \left(2 \cdot \left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot 0.008333333333333333}{\ell} - 0.3333333333333333 \cdot \frac{\left(t\_m \cdot t\_m\right) \cdot -0.5}{\ell}\right) + t\_2 \cdot -0.041666666666666664\right) + \left(-0.3333333333333333 \cdot \left(t\_2 \cdot 0.3333333333333333\right) + \frac{1}{\ell} \cdot -0.16666666666666666\right)\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m t_m) l)))
   (*
    t_s
    (if (<= t_m 1.06e-38)
      (/
       2.0
       (/
        (*
         (* k k)
         (+
          (/ (* 2.0 (* t_m (* t_m t_m))) l)
          (*
           (* k k)
           (+
            (*
             t_m
             (+
              (* (/ (* 2.0 (* t_m t_m)) l) 0.3333333333333333)
              (+ (/ 1.0 l) (/ (* (* t_m t_m) -0.3333333333333333) l))))
            (*
             (+
              (+
               (/ (* (* t_m t_m) 0.016666666666666666) l)
               (/ 0.3333333333333333 l))
              (+
               (*
                2.0
                (+
                 (-
                  (/ (* (* t_m t_m) 0.008333333333333333) l)
                  (* 0.3333333333333333 (/ (* (* t_m t_m) -0.5) l)))
                 (* t_2 -0.041666666666666664)))
               (+
                (* -0.3333333333333333 (* t_2 0.3333333333333333))
                (* (/ 1.0 l) -0.16666666666666666))))
             (* t_m (* k k)))))))
        l))
      (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 1.06e-38) {
		tmp = 2.0 / (((k * k) * (((2.0 * (t_m * (t_m * t_m))) / l) + ((k * k) * ((t_m * ((((2.0 * (t_m * t_m)) / l) * 0.3333333333333333) + ((1.0 / l) + (((t_m * t_m) * -0.3333333333333333) / l)))) + ((((((t_m * t_m) * 0.016666666666666666) / l) + (0.3333333333333333 / l)) + ((2.0 * (((((t_m * t_m) * 0.008333333333333333) / l) - (0.3333333333333333 * (((t_m * t_m) * -0.5) / l))) + (t_2 * -0.041666666666666664))) + ((-0.3333333333333333 * (t_2 * 0.3333333333333333)) + ((1.0 / l) * -0.16666666666666666)))) * (t_m * (k * k))))))) / l);
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m * t_m) / l
    if (t_m <= 1.06d-38) then
        tmp = 2.0d0 / (((k * k) * (((2.0d0 * (t_m * (t_m * t_m))) / l) + ((k * k) * ((t_m * ((((2.0d0 * (t_m * t_m)) / l) * 0.3333333333333333d0) + ((1.0d0 / l) + (((t_m * t_m) * (-0.3333333333333333d0)) / l)))) + ((((((t_m * t_m) * 0.016666666666666666d0) / l) + (0.3333333333333333d0 / l)) + ((2.0d0 * (((((t_m * t_m) * 0.008333333333333333d0) / l) - (0.3333333333333333d0 * (((t_m * t_m) * (-0.5d0)) / l))) + (t_2 * (-0.041666666666666664d0)))) + (((-0.3333333333333333d0) * (t_2 * 0.3333333333333333d0)) + ((1.0d0 / l) * (-0.16666666666666666d0))))) * (t_m * (k * k))))))) / l)
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 1.06e-38) {
		tmp = 2.0 / (((k * k) * (((2.0 * (t_m * (t_m * t_m))) / l) + ((k * k) * ((t_m * ((((2.0 * (t_m * t_m)) / l) * 0.3333333333333333) + ((1.0 / l) + (((t_m * t_m) * -0.3333333333333333) / l)))) + ((((((t_m * t_m) * 0.016666666666666666) / l) + (0.3333333333333333 / l)) + ((2.0 * (((((t_m * t_m) * 0.008333333333333333) / l) - (0.3333333333333333 * (((t_m * t_m) * -0.5) / l))) + (t_2 * -0.041666666666666664))) + ((-0.3333333333333333 * (t_2 * 0.3333333333333333)) + ((1.0 / l) * -0.16666666666666666)))) * (t_m * (k * k))))))) / l);
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m * t_m) / l
	tmp = 0
	if t_m <= 1.06e-38:
		tmp = 2.0 / (((k * k) * (((2.0 * (t_m * (t_m * t_m))) / l) + ((k * k) * ((t_m * ((((2.0 * (t_m * t_m)) / l) * 0.3333333333333333) + ((1.0 / l) + (((t_m * t_m) * -0.3333333333333333) / l)))) + ((((((t_m * t_m) * 0.016666666666666666) / l) + (0.3333333333333333 / l)) + ((2.0 * (((((t_m * t_m) * 0.008333333333333333) / l) - (0.3333333333333333 * (((t_m * t_m) * -0.5) / l))) + (t_2 * -0.041666666666666664))) + ((-0.3333333333333333 * (t_2 * 0.3333333333333333)) + ((1.0 / l) * -0.16666666666666666)))) * (t_m * (k * k))))))) / l)
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 1.06e-38)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(Float64(2.0 * Float64(t_m * Float64(t_m * t_m))) / l) + Float64(Float64(k * k) * Float64(Float64(t_m * Float64(Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) / l) * 0.3333333333333333) + Float64(Float64(1.0 / l) + Float64(Float64(Float64(t_m * t_m) * -0.3333333333333333) / l)))) + Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * 0.016666666666666666) / l) + Float64(0.3333333333333333 / l)) + Float64(Float64(2.0 * Float64(Float64(Float64(Float64(Float64(t_m * t_m) * 0.008333333333333333) / l) - Float64(0.3333333333333333 * Float64(Float64(Float64(t_m * t_m) * -0.5) / l))) + Float64(t_2 * -0.041666666666666664))) + Float64(Float64(-0.3333333333333333 * Float64(t_2 * 0.3333333333333333)) + Float64(Float64(1.0 / l) * -0.16666666666666666)))) * Float64(t_m * Float64(k * k))))))) / l));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m * t_m) / l;
	tmp = 0.0;
	if (t_m <= 1.06e-38)
		tmp = 2.0 / (((k * k) * (((2.0 * (t_m * (t_m * t_m))) / l) + ((k * k) * ((t_m * ((((2.0 * (t_m * t_m)) / l) * 0.3333333333333333) + ((1.0 / l) + (((t_m * t_m) * -0.3333333333333333) / l)))) + ((((((t_m * t_m) * 0.016666666666666666) / l) + (0.3333333333333333 / l)) + ((2.0 * (((((t_m * t_m) * 0.008333333333333333) / l) - (0.3333333333333333 * (((t_m * t_m) * -0.5) / l))) + (t_2 * -0.041666666666666664))) + ((-0.3333333333333333 * (t_2 * 0.3333333333333333)) + ((1.0 / l) * -0.16666666666666666)))) * (t_m * (k * k))))))) / l);
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.06e-38], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(2.0 * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[(1.0 / l), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.016666666666666666), $MachinePrecision] / l), $MachinePrecision] + N[(0.3333333333333333 / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] / l), $MachinePrecision] - N[(0.3333333333333333 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[(t$95$2 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / l), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.06000000000000001e-38

   1. Initial program 58.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 56.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 64.2%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\sin k \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \ell\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \sin k\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \sin k\right), \ell\right)\right) \]

   8. Applied egg-rr 51.4%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)}, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({t}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \sin k\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       15. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

       17. cos-lowering-cos.f64 84.6%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right) \]

   11. Simplified 84.6%

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

   12. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{\ell} + {k}^{2} \cdot \left(t \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} + \left(2 \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right) + {k}^{2} \cdot \left(t \cdot \left(\frac{-1}{6} \cdot \left(2 \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right) + \left(\frac{1}{60} \cdot \frac{{t}^{2}}{\ell} + \left(\frac{1}{3} \cdot \frac{1}{\ell} + 2 \cdot \left(\frac{1}{120} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{2} \cdot \left(\frac{-1}{6} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right)\right)\right)\right)\right)\right)\right)\right)}, \ell\right)\right) \]

   14. Simplified 57.2%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell} + \left(k \cdot k\right) \cdot \left(t \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} \cdot 0.3333333333333333 + \left(\frac{1}{\ell} + \frac{-0.3333333333333333 \cdot \left(t \cdot t\right)}{\ell}\right)\right) + \left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{0.016666666666666666 \cdot \left(t \cdot t\right)}{\ell} + \frac{0.3333333333333333}{\ell}\right) + \left(2 \cdot \left(\left(\frac{\left(t \cdot t\right) \cdot 0.008333333333333333}{\ell} - \frac{-0.5 \cdot \left(t \cdot t\right)}{\ell} \cdot 0.3333333333333333\right) + -0.041666666666666664 \cdot \frac{t \cdot t}{\ell}\right) + \left(-0.3333333333333333 \cdot \left(\frac{t \cdot t}{\ell} \cdot 0.3333333333333333\right) + -0.16666666666666666 \cdot \frac{1}{\ell}\right)\right)\right)\right)\right)}}{\ell}} \]

   if 1.06000000000000001e-38 < t

   1. Initial program 59.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.1%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 79.1%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.06 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell} + \left(k \cdot k\right) \cdot \left(t \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} \cdot 0.3333333333333333 + \left(\frac{1}{\ell} + \frac{\left(t \cdot t\right) \cdot -0.3333333333333333}{\ell}\right)\right) + \left(\left(\frac{\left(t \cdot t\right) \cdot 0.016666666666666666}{\ell} + \frac{0.3333333333333333}{\ell}\right) + \left(2 \cdot \left(\left(\frac{\left(t \cdot t\right) \cdot 0.008333333333333333}{\ell} - 0.3333333333333333 \cdot \frac{\left(t \cdot t\right) \cdot -0.5}{\ell}\right) + \frac{t \cdot t}{\ell} \cdot -0.041666666666666664\right) + \left(-0.3333333333333333 \cdot \left(\frac{t \cdot t}{\ell} \cdot 0.3333333333333333\right) + \frac{1}{\ell} \cdot -0.16666666666666666\right)\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-39)
    (/ 2.0 (* t_m (/ (/ (* k (* k (* k k))) l) l)))
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = 2.0 / (t_m * (((k * (k * (k * k))) / l) / l));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.75d-39) then
        tmp = 2.0d0 / (t_m * (((k * (k * (k * k))) / l) / l))
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = 2.0 / (t_m * (((k * (k * (k * k))) / l) / l));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.75e-39:
		tmp = 2.0 / (t_m * (((k * (k * (k * k))) / l) / l))
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-39)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(k * Float64(k * Float64(k * k))) / l) / l)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.75e-39)
		tmp = 2.0 / (t_m * (((k * (k * (k * k))) / l) / l));
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-39], N[(2.0 / N[(t$95$m * N[(N[(N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.75000000000000009e-39

   1. Initial program 58.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 56.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 64.2%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right)}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{1}{\ell}\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3} \cdot 1}{\ell}\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{\ell}\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \left(\ell \cdot {t}^{2}\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \left({t}^{2}\right)\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(t \cdot t\right)\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \left(\frac{2 \cdot 1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \left(\frac{2}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      18. /-lowering-/.f64 42.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

   9. Simplified 42.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{0.3333333333333333}{\ell} + \frac{1}{\ell \cdot \left(t \cdot t\right)}\right) + \frac{2}{\ell}\right)\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\color{blue}{\left(\frac{{k}^{4} \cdot t}{\ell}\right)}, \ell\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \ell\right), \ell\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \ell\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \ell\right), \ell\right)\right) \]

       4. pow-lowering-pow.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \ell\right), \ell\right)\right) \]

   12. Simplified 62.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t \cdot {k}^{4}}{\ell}}}{\ell}} \]
   13. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \frac{{k}^{4}}{\ell}}{\ell}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(t \cdot \color{blue}{\frac{\frac{{k}^{4}}{\ell}}{\ell}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{\frac{{k}^{4}}{\ell}}{\ell}\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\left(\frac{{k}^{4}}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{4}\right), \ell\right), \ell\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \ell\right), \ell\right)\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \ell\right), \ell\right)\right)\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot {k}^{2}\right), \ell\right), \ell\right)\right)\right) \]

       9. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right), \ell\right), \ell\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right), \ell\right)\right)\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(k \cdot {k}^{3}\right), \ell\right), \ell\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3}\right)\right), \ell\right), \ell\right)\right)\right) \]

       13. cube-unmult N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right), \ell\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right), \ell\right)\right)\right) \]

       15. *-lowering-*.f64 63.7%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right), \ell\right)\right)\right) \]

   14. Applied egg-rr 63.7%

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

   if 2.75000000000000009e-39 < t

   1. Initial program 59.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.1%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 79.1%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 75.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m}}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-39)
    (* l (/ (/ 2.0 t_m) (/ (* k (* k (* k k))) l)))
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = l * ((2.0 / t_m) / ((k * (k * (k * k))) / l));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.75d-39) then
        tmp = l * ((2.0d0 / t_m) / ((k * (k * (k * k))) / l))
    else
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-39) {
		tmp = l * ((2.0 / t_m) / ((k * (k * (k * k))) / l));
	} else {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.75e-39:
		tmp = l * ((2.0 / t_m) / ((k * (k * (k * k))) / l))
	else:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-39)
		tmp = Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(k * Float64(k * Float64(k * k))) / l)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.75e-39)
		tmp = l * ((2.0 / t_m) / ((k * (k * (k * k))) / l));
	else
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-39], N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.75000000000000009e-39

   1. Initial program 58.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 56.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell}}{\ell} \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   6. Applied egg-rr 64.2%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right)}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right) + 2 \cdot \frac{1}{\ell}\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{3} \cdot \frac{1}{\ell} + \frac{1}{\ell \cdot {t}^{2}}\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{1}{\ell}\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3} \cdot 1}{\ell}\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{\ell}\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \left(\frac{1}{\ell \cdot {t}^{2}}\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \left(\ell \cdot {t}^{2}\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \left({t}^{2}\right)\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(t \cdot t\right)\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \left(2 \cdot \frac{1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \left(\frac{2 \cdot 1}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \left(\frac{2}{\ell}\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

      18. /-lowering-/.f64 42.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \ell\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), \ell\right)\right) \]

   9. Simplified 42.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{0.3333333333333333}{\ell} + \frac{1}{\ell \cdot \left(t \cdot t\right)}\right) + \frac{2}{\ell}\right)\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\color{blue}{\left(\frac{{k}^{4} \cdot t}{\ell}\right)}, \ell\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \ell\right), \ell\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \ell\right), \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \ell\right), \ell\right)\right) \]

       4. pow-lowering-pow.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \ell\right), \ell\right)\right) \]

   12. Simplified 62.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t \cdot {k}^{4}}{\ell}}}{\ell}} \]
   13. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \cdot \color{blue}{\ell} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\frac{t \cdot {k}^{4}}{\ell}}\right), \color{blue}{\ell}\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \frac{{k}^{4}}{\ell}}\right), \ell\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t}}{\frac{{k}^{4}}{\ell}}\right), \ell\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{{k}^{4}}{\ell}\right)\right), \ell\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{{k}^{4}}{\ell}\right)\right), \ell\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left({k}^{4}\right), \ell\right)\right), \ell\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \ell\right)\right), \ell\right) \]

       9. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \ell\right)\right), \ell\right) \]

       10. pow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot {k}^{2}\right), \ell\right)\right), \ell\right) \]

       11. pow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right), \ell\right)\right), \ell\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right), \ell\right) \]

       13. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot {k}^{3}\right), \ell\right)\right), \ell\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left({k}^{3}\right)\right), \ell\right)\right), \ell\right) \]

       15. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right), \ell\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right)\right), \ell\right) \]

       17. *-lowering-*.f64 63.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right)\right), \ell\right) \]

   14. Applied egg-rr 63.6%

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

   if 2.75000000000000009e-39 < t

   1. Initial program 59.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.1%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 79.1%

       \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t}}{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.0% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t\_m \cdot \left(k \cdot k\right)}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.55e+78)
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))
    (/ (/ (/ (* l l) (* t_m (* k k))) t_m) 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 (k <= 1.55e+78) {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	} else {
		tmp = (((l * l) / (t_m * (k * k))) / t_m) / 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 (k <= 1.55d+78) then
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    else
        tmp = (((l * l) / (t_m * (k * k))) / t_m) / 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 (k <= 1.55e+78) {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	} else {
		tmp = (((l * l) / (t_m * (k * k))) / t_m) / 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 k <= 1.55e+78:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	else:
		tmp = (((l * l) / (t_m * (k * k))) / t_m) / 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 (k <= 1.55e+78)
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(t_m * Float64(k * k))) / t_m) / 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 (k <= 1.55e+78)
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	else
		tmp = (((l * l) / (t_m * (k * k))) / t_m) / 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[k, 1.55e+78], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55e78

   1. Initial program 58.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 58.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 64.3%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 70.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 70.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 74.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 74.7%

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

   if 1.55e78 < k

   1. Initial program 60.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 60.5%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 59.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.5%

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

       1. associate-*r/ N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k \cdot k} \cdot \ell}{t}\right), \color{blue}{t}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \ell\right), t\right), t\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell\right), t\right), t\right) \]

       6. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}\right), t\right), t\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\left(k \cdot k\right) \cdot t\right)\right), t\right), t\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot t\right)\right), t\right), t\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(t \cdot \left(k \cdot k\right)\right)\right), t\right), t\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), t\right), t\right) \]

       11. *-lowering-*.f64 69.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), t\right), t\right) \]

   11. Applied egg-rr 69.6%

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

Alternative 19: 72.2% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{+62}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(k \cdot k\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e+62)
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))
    (/ (* (/ l t_m) (/ l (* t_m (* 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 (k <= 1e+62) {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	} else {
		tmp = ((l / t_m) * (l / (t_m * (k * 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 (k <= 1d+62) then
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    else
        tmp = ((l / t_m) * (l / (t_m * (k * 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 (k <= 1e+62) {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	} else {
		tmp = ((l / t_m) * (l / (t_m * (k * 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 k <= 1e+62:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	else:
		tmp = ((l / t_m) * (l / (t_m * (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 (k <= 1e+62)
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(k * 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 (k <= 1e+62)
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	else
		tmp = ((l / t_m) * (l / (t_m * (k * 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[k, 1e+62], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000004e62

   1. Initial program 58.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.2%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 58.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 64.4%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 70.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 70.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 75.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 75.0%

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

   if 1.00000000000000004e62 < k

   1. Initial program 62.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 62.2%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 59.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 59.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 59.4%

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

       1. associate-/r* N/A

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

       2. associate-*r/ N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{t}\right), \color{blue}{t}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\ell}{t}\right)\right), t\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \left(\frac{\ell}{t}\right)\right), t\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot t\right)\right), \left(\frac{\ell}{t}\right)\right), t\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\ell}{t}\right)\right), t\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \left(\frac{\ell}{t}\right)\right), t\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\frac{\ell}{t}\right)\right), t\right) \]

       10. /-lowering-/.f64 65.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right) \]

   11. Applied egg-rr 65.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+62}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 71.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t\_m}}{k \cdot k}}{t\_m \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e+24)
    (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))
    (/ (* l (/ (/ l t_m) (* k k))) (* t_m 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 (k <= 5e+24) {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	} else {
		tmp = (l * ((l / t_m) / (k * k))) / (t_m * 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 (k <= 5d+24) then
        tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
    else
        tmp = (l * ((l / t_m) / (k * k))) / (t_m * 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 (k <= 5e+24) {
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	} else {
		tmp = (l * ((l / t_m) / (k * k))) / (t_m * 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 k <= 5e+24:
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k))
	else:
		tmp = (l * ((l / t_m) / (k * k))) / (t_m * 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 (k <= 5e+24)
		tmp = Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	else
		tmp = Float64(Float64(l * Float64(Float64(l / t_m) / Float64(k * k))) / Float64(t_m * 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 (k <= 5e+24)
		tmp = ((l / t_m) / k) * ((l / t_m) / (t_m * k));
	else
		tmp = (l * ((l / t_m) / (k * k))) / (t_m * 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[k, 5e+24], N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(l / t$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000045e24

   1. Initial program 58.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.3%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 59.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

   9. Applied egg-rr 65.3%

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

       1. associate-*l/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

       8. *-lowering-*.f64 71.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

   11. Applied egg-rr 71.4%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       6. *-lowering-*.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   13. Applied egg-rr 76.2%

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

   if 5.00000000000000045e24 < k

   1. Initial program 60.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 60.9%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 56.3%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(t \cdot t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)\right), \left(\color{blue}{t} \cdot t\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\right), \left(t \cdot t\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{k \cdot k}\right)\right), \left(t \cdot t\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right)\right), \left(t \cdot t\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right)\right), \left(t \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(t \cdot t\right)\right) \]

      10. *-lowering-*.f64 56.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot k}}{t \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 70.9% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\frac{\ell}{t\_m}}{k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ (/ l t_m) k) (/ (/ l t_m) (* t_m k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) / k) * ((l / t_m) / (t_m * k)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / t_m) / k) * ((l / t_m) / (t_m * k)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) / k) * ((l / t_m) / (t_m * k)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l / t_m) / k) * ((l / t_m) / (t_m * k)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l / t_m) / k) * Float64(Float64(l / t_m) / Float64(t_m * k))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / t_m) / k) * ((l / t_m) / (t_m * k)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

3. Simplified 55.8%

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

5. Taylor expanded in k around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. unpow3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

   14. *-lowering-*.f64 58.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

7. Simplified 58.5%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   10. *-lowering-*.f64 63.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

9. Applied egg-rr 63.4%

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

    1. associate-*l/ N/A

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

    2. times-frac N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

    8. *-lowering-*.f64 67.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

11. Applied egg-rr 67.9%

    \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
12. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\frac{\ell}{t}}{t}}{k}\right)\right) \]

    2. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot t}}\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \color{blue}{k}\right)\right)\right) \]

    6. *-lowering-*.f64 71.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

13. Applied egg-rr 71.6%

    \[\leadsto \frac{\frac{\ell}{t}}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]
14. Add Preprocessing

Alternative 22: 64.0% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{k}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ (/ l k) t_m) (/ (/ l (* t_m t_m)) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / k) / t_m) * ((l / (t_m * t_m)) / k));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / k) / t_m) * ((l / (t_m * t_m)) / k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / k) / t_m) * ((l / (t_m * t_m)) / k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l / k) / t_m) * ((l / (t_m * t_m)) / k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l / k) / t_m) * Float64(Float64(l / Float64(t_m * t_m)) / k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / k) / t_m) * ((l / (t_m * t_m)) / k));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

3. Simplified 55.8%

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

5. Taylor expanded in k around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. unpow3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

   14. *-lowering-*.f64 58.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

7. Simplified 58.5%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot t}\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot t}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot t}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot t\right)}\right)\right) \]

   10. *-lowering-*.f64 63.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right) \]

9. Applied egg-rr 63.4%

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

    1. associate-*l/ N/A

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

    2. times-frac N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t \cdot t}}{k}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(\frac{\frac{\color{blue}{\ell}}{t \cdot t}}{k}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{k}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), k\right)\right) \]

    8. *-lowering-*.f64 67.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

11. Applied egg-rr 67.9%

    \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}} \]
12. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)}, k\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k}}{t}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)}, k\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), t\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)}, k\right)\right) \]

    4. /-lowering-/.f64 68.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right) \]

13. Applied egg-rr 68.2%

    \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t}} \cdot \frac{\frac{\ell}{t \cdot t}}{k} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(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))))