Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 84.1%

Time: 18.0s

Alternatives: 13

Speedup: 32.4×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(t\_m \cdot \frac{t\_m \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 3e-87)
    (/ 2.0 (* k (* k (* (/ t_m (* l l)) (* (sin k) (tan k))))))
    (if (<= t_m 3.5e+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
       (*
        (* (tan k) (* (sin k) (* t_m (/ (* t_m (/ t_m l)) l))))
        (+ 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 <= 3e-87) {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	} else if (t_m <= 3.5e+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 / ((tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3d-87) then
        tmp = 2.0d0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))))
    else if (t_m <= 3.5d+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 / ((tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-87) {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (Math.sin(k) * Math.tan(k)))));
	} else if (t_m <= 3.5e+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 / ((Math.tan(k) * (Math.sin(k) * (t_m * ((t_m * (t_m / l)) / l)))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3e-87:
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (math.sin(k) * math.tan(k)))))
	elif t_m <= 3.5e+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 / ((math.tan(k) * (math.sin(k) * (t_m * ((t_m * (t_m / l)) / l)))) * (1.0 + (1.0 + math.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 <= 3e-87)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_m / Float64(l * l)) * Float64(sin(k) * tan(k))))));
	elseif (t_m <= 3.5e+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(tan(k) * Float64(sin(k) * Float64(t_m * Float64(Float64(t_m * Float64(t_m / l)) / l)))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3e-87)
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	elseif (t_m <= 3.5e+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 / ((tan(k) * (sin(k) * (t_m * ((t_m * (t_m / l)) / l)))) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-87], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+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[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $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 3 regimes

2. if t < 3.00000000000000016e-87

   1. Initial program 51.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.6%

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

   5. 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. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. cos-lowering-cos.f64 64.9%

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

   7. Simplified 64.9%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. tan-quot N/A

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

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

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

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

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 69.1%

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

   9. Applied egg-rr 69.1%

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

   if 3.00000000000000016e-87 < t < 3.50000000000000011e102

   1. Initial program 72.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   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 89.1%

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

   if 3.50000000000000011e102 < t

   1. Initial program 62.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 83.5%

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

   4. Applied egg-rr 83.5%

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

      1. unpow2 N/A

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

      2. frac-times N/A

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

      3. pow-prod-up N/A

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

      4. metadata-eval N/A

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

      5. cube-unmult N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/r* N/A

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

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

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

      11. associate-/l* N/A

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

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

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

      13. /-lowering-/.f64 88.4%

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

   6. Applied egg-rr 88.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \left(t\_m \cdot t\_m\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

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 3e-87)
    (/ 2.0 (* k (* k (* (/ t_m (* l l)) (* (sin k) (tan k))))))
    (if (<= t_m 3e+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
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (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 <= 3e-87) {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	} else if (t_m <= 3e+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 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (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 <= 3d-87) then
        tmp = 2.0d0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))))
    else if (t_m <= 3d+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 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (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 <= 3e-87) {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (Math.sin(k) * Math.tan(k)))));
	} else if (t_m <= 3e+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 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (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 <= 3e-87:
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (math.sin(k) * math.tan(k)))))
	elif t_m <= 3e+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 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (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 <= 3e-87)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_m / Float64(l * l)) * Float64(sin(k) * tan(k))))));
	elseif (t_m <= 3e+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(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * 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 <= 3e-87)
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	elseif (t_m <= 3e+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 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (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, 3e-87], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+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[(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[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.00000000000000016e-87

   1. Initial program 51.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.6%

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

   5. 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. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. cos-lowering-cos.f64 64.9%

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

   7. Simplified 64.9%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. tan-quot N/A

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

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

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

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

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 69.1%

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

   9. Applied egg-rr 69.1%

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

   if 3.00000000000000016e-87 < t < 2.9999999999999998e102

   1. Initial program 72.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   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 89.1%

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

   if 2.9999999999999998e102 < t

   1. Initial program 62.9%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 78.6%

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

   4. Applied egg-rr 78.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\frac{\sin k}{\ell}}}{\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 0.00066:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{1}{k \cdot k}}{t\_m \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)}}{\frac{t\_m \cdot k}{\ell \cdot \ell}}}{k}}{\frac{1}{\cos k}}\\ \end{array} \end{array} \]

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 (<= k 1.3e-155)
    (* (/ l k) (/ l (* t_m (* t_m (* t_m k)))))
    (if (<= k 0.00066)
      (* (/ l t_m) (/ (/ 1.0 (* k k)) (* t_m (/ t_m l))))
      (/
       (/ (/ (/ 2.0 (+ 0.5 (* -0.5 (cos (* 2.0 k))))) (/ (* t_m k) (* l l))) k)
       (/ 1.0 (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 0.00066) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (((2.0 / (0.5 + (-0.5 * cos((2.0 * k))))) / ((t_m * k) / (l * l))) / k) / (1.0 / cos(k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-155) then
        tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
    else if (k <= 0.00066d0) then
        tmp = (l / t_m) * ((1.0d0 / (k * k)) / (t_m * (t_m / l)))
    else
        tmp = (((2.0d0 / (0.5d0 + ((-0.5d0) * cos((2.0d0 * k))))) / ((t_m * k) / (l * l))) / k) / (1.0d0 / cos(k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 0.00066) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (((2.0 / (0.5 + (-0.5 * Math.cos((2.0 * k))))) / ((t_m * k) / (l * l))) / k) / (1.0 / Math.cos(k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.3e-155:
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
	elif k <= 0.00066:
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)))
	else:
		tmp = (((2.0 / (0.5 + (-0.5 * math.cos((2.0 * k))))) / ((t_m * k) / (l * l))) / k) / (1.0 / math.cos(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.3e-155)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * k)))));
	elseif (k <= 0.00066)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(1.0 / Float64(k * k)) / Float64(t_m * Float64(t_m / l))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * k))))) / Float64(Float64(t_m * k) / Float64(l * l))) / k) / Float64(1.0 / cos(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.3e-155)
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	elseif (k <= 0.00066)
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	else
		tmp = (((2.0 / (0.5 + (-0.5 * cos((2.0 * k))))) / ((t_m * k) / (l * l))) / k) / (1.0 / cos(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.3e-155], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.00066], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(0.5 + N[(-0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(1.0 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4%

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

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

      5. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 50.1%

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

   7. Simplified 50.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.6%

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

   9. Applied egg-rr 67.6%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 71.6%

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

   11. Applied egg-rr 71.6%

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

   if 1.30000000000000004e-155 < k < 6.6e-4

   1. Initial program 65.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 23.4%

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

   4. Applied egg-rr 23.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{{k}^{2}}}{\color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({k}^{2}\right)\right), \left(e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      6. exp-diff N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right)\right) \]

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      18. *-lowering-*.f64 68.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 68.8%

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

      1. div-inv N/A

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

      2. times-frac N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot t}{\ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}\right)}\right) \]

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 90.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}} \]

   if 6.6e-4 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.5%

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

   5. Taylor expanded in 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. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. cos-lowering-cos.f64 67.9%

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

   7. Simplified 67.9%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)}}{\frac{1}{\cos k}}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}{\frac{t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{cos.f64}\left(k\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}{\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k}\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}{\frac{t}{\ell \cdot \ell} \cdot k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{cos.f64}\left(k\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}{\frac{t}{\ell \cdot \ell} \cdot k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{cos.f64}\left(k\right)\right)\right) \]

   11. Applied egg-rr 76.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{2}{0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)}}{\frac{t \cdot k}{\ell \cdot \ell}}}{k}}}{\frac{1}{\cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 0.00066:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)}}{\frac{t \cdot k}{\ell \cdot \ell}}}{k}}{\frac{1}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 0.0031:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{1}{k \cdot k}}{t\_m \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(k \cdot \frac{t\_m \cdot k}{\ell \cdot \ell}\right)}}{\frac{1}{\cos k}}\\ \end{array} \end{array} \]

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 (<= k 1.3e-155)
    (* (/ l k) (/ l (* t_m (* t_m (* t_m k)))))
    (if (<= k 0.0031)
      (* (/ l t_m) (/ (/ 1.0 (* k k)) (* t_m (/ t_m l))))
      (/
       (/ 2.0 (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) (* k (/ (* t_m k) (* l l)))))
       (/ 1.0 (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 0.0031) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * (k * ((t_m * k) / (l * l))))) / (1.0 / cos(k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-155) then
        tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
    else if (k <= 0.0031d0) then
        tmp = (l / t_m) * ((1.0d0 / (k * k)) / (t_m * (t_m / l)))
    else
        tmp = (2.0d0 / ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * (k * ((t_m * k) / (l * l))))) / (1.0d0 / cos(k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 0.0031) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (2.0 / ((0.5 - (0.5 * Math.cos((2.0 * k)))) * (k * ((t_m * k) / (l * l))))) / (1.0 / Math.cos(k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.3e-155:
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
	elif k <= 0.0031:
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)))
	else:
		tmp = (2.0 / ((0.5 - (0.5 * math.cos((2.0 * k)))) * (k * ((t_m * k) / (l * l))))) / (1.0 / math.cos(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.3e-155)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * k)))));
	elseif (k <= 0.0031)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(1.0 / Float64(k * k)) / Float64(t_m * Float64(t_m / l))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * Float64(k * Float64(Float64(t_m * k) / Float64(l * l))))) / Float64(1.0 / cos(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.3e-155)
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	elseif (k <= 0.0031)
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	else
		tmp = (2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * (k * ((t_m * k) / (l * l))))) / (1.0 / cos(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.3e-155], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0031], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(t$95$m * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4%

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

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

      5. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 50.1%

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

   7. Simplified 50.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.6%

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

   9. Applied egg-rr 67.6%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 71.6%

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

   11. Applied egg-rr 71.6%

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

   if 1.30000000000000004e-155 < k < 0.00309999999999999989

   1. Initial program 65.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 23.4%

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

   4. Applied egg-rr 23.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{{k}^{2}}}{\color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({k}^{2}\right)\right), \left(e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      6. exp-diff N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right)\right) \]

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      18. *-lowering-*.f64 68.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 68.8%

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

      1. div-inv N/A

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

      2. times-frac N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot t}{\ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}\right)}\right) \]

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 90.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}} \]

   if 0.00309999999999999989 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.5%

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

   5. Taylor expanded in 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. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. cos-lowering-cos.f64 67.9%

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

   7. Simplified 67.9%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

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

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

   9. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)}}{\frac{1}{\cos k}}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), \left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right), k\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{t \cdot k}{\ell \cdot \ell}\right), k\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(t \cdot k\right), \left(\ell \cdot \ell\right)\right), k\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(\ell \cdot \ell\right)\right), k\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

       6. *-lowering-*.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), k\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(k\right)\right)\right) \]

   11. Applied egg-rr 76.6%

       \[\leadsto \frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \color{blue}{\left(\frac{t \cdot k}{\ell \cdot \ell} \cdot k\right)}}}{\frac{1}{\cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 0.0031:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(k \cdot \frac{t \cdot k}{\ell \cdot \ell}\right)}}{\frac{1}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-124}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \sin k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(\tan k \cdot \frac{2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}}{\ell}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

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 7e-124)
    (/ 2.0 (* k (* k (* (/ t_m (* l l)) (* (sin k) (tan k))))))
    (/
     2.0
     (/
      (*
       (* t_m (sin k))
       (* (* t_m t_m) (* (tan k) (/ (+ 2.0 (/ k (/ (* t_m t_m) k))) l))))
      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 <= 7e-124) {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	} else {
		tmp = 2.0 / (((t_m * sin(k)) * ((t_m * t_m) * (tan(k) * ((2.0 + (k / ((t_m * t_m) / k))) / l)))) / 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 <= 7d-124) then
        tmp = 2.0d0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))))
    else
        tmp = 2.0d0 / (((t_m * sin(k)) * ((t_m * t_m) * (tan(k) * ((2.0d0 + (k / ((t_m * t_m) / k))) / l)))) / 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 <= 7e-124) {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (Math.sin(k) * Math.tan(k)))));
	} else {
		tmp = 2.0 / (((t_m * Math.sin(k)) * ((t_m * t_m) * (Math.tan(k) * ((2.0 + (k / ((t_m * t_m) / k))) / l)))) / 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 <= 7e-124:
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (math.sin(k) * math.tan(k)))))
	else:
		tmp = 2.0 / (((t_m * math.sin(k)) * ((t_m * t_m) * (math.tan(k) * ((2.0 + (k / ((t_m * t_m) / k))) / l)))) / 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 <= 7e-124)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_m / Float64(l * l)) * Float64(sin(k) * tan(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(Float64(t_m * t_m) * Float64(tan(k) * Float64(Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k))) / l)))) / 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 <= 7e-124)
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	else
		tmp = 2.0 / (((t_m * sin(k)) * ((t_m * t_m) * (tan(k) * ((2.0 + (k / ((t_m * t_m) / k))) / l)))) / 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, 7e-124], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.9999999999999997e-124

   1. Initial program 50.1%

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

      1. /-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 46.6%

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

   5. 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. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. cos-lowering-cos.f64 64.4%

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

   7. Simplified 64.4%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. tan-quot N/A

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

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

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

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

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 68.7%

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

   9. Applied egg-rr 68.7%

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

   if 6.9999999999999997e-124 < t

   1. Initial program 68.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 60.1%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 70.1%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

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

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

   8. Applied egg-rr 76.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-/r/ N/A

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

      13. associate-*l/ N/A

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

      14. frac-times N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. associate-+r+ N/A

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

      18. +-commutative N/A

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

   10. Applied egg-rr 82.9%

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

Alternative 6: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 0.0165:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{1}{k \cdot k}}{t\_m \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 1.3e-155)
    (* (/ l k) (/ l (* t_m (* t_m (* t_m k)))))
    (if (<= k 0.0165)
      (* (/ l t_m) (/ (/ 1.0 (* k k)) (* t_m (/ t_m l))))
      (/ 2.0 (* k (* k (* (/ t_m (* l l)) (* (sin k) (tan k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 0.0165) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-155) then
        tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
    else if (k <= 0.0165d0) then
        tmp = (l / t_m) * ((1.0d0 / (k * k)) / (t_m * (t_m / l)))
    else
        tmp = 2.0d0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 0.0165) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (Math.sin(k) * Math.tan(k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.3e-155:
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
	elif k <= 0.0165:
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)))
	else:
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (math.sin(k) * math.tan(k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.3e-155)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * k)))));
	elseif (k <= 0.0165)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(1.0 / Float64(k * k)) / Float64(t_m * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_m / Float64(l * l)) * Float64(sin(k) * tan(k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.3e-155)
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	elseif (k <= 0.0165)
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	else
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (sin(k) * tan(k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.3e-155], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0165], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4%

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

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

      5. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 50.1%

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

   7. Simplified 50.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.6%

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

   9. Applied egg-rr 67.6%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 71.6%

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

   11. Applied egg-rr 71.6%

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

   if 1.30000000000000004e-155 < k < 0.016500000000000001

   1. Initial program 65.9%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 23.4%

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

   4. Applied egg-rr 23.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{{k}^{2}}}{\color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({k}^{2}\right)\right), \left(e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      6. exp-diff N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right)\right) \]

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      18. *-lowering-*.f64 68.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 68.8%

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

      1. div-inv N/A

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

      2. times-frac N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot t}{\ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}\right)}\right) \]

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 90.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}} \]

   if 0.016500000000000001 < k

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.5%

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

   5. Taylor expanded in 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. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. cos-lowering-cos.f64 67.9%

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

   7. Simplified 67.9%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. tan-quot N/A

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

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

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

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

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 73.9%

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

   9. Applied egg-rr 73.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 0.0165:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 10.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{1}{k \cdot k}}{t\_m \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(2 \cdot \left(t\_m \cdot t\_m\right) + k \cdot \left(k \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 1.3e-155)
    (* (/ l k) (/ l (* t_m (* t_m (* t_m k)))))
    (if (<= k 1.2e-25)
      (* (/ l t_m) (/ (/ 1.0 (* k k)) (* t_m (/ t_m l))))
      (/
       (* 2.0 (* l l))
       (*
        (+
         (* 2.0 (* t_m t_m))
         (* k (* k (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))))
        (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 1.2e-25) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (2.0 * (l * l)) / (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-155) then
        tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
    else if (k <= 1.2d-25) then
        tmp = (l / t_m) * ((1.0d0 / (k * k)) / (t_m * (t_m / l)))
    else
        tmp = (2.0d0 * (l * l)) / (((2.0d0 * (t_m * t_m)) + (k * (k * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))))) * (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 1.2e-25) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (2.0 * (l * l)) / (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.3e-155:
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
	elif k <= 1.2e-25:
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)))
	else:
		tmp = (2.0 * (l * l)) / (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) * (t_m * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.3e-155)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * k)))));
	elseif (k <= 1.2e-25)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(1.0 / Float64(k * k)) / Float64(t_m * Float64(t_m / l))));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(k * Float64(k * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))))) * Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.3e-155)
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	elseif (k <= 1.2e-25)
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	else
		tmp = (2.0 * (l * l)) / (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) * (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.3e-155], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e-25], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(k * N[(k * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4%

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

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

      5. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 50.1%

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

   7. Simplified 50.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.6%

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

   9. Applied egg-rr 67.6%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 71.6%

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

   11. Applied egg-rr 71.6%

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

   if 1.30000000000000004e-155 < k < 1.20000000000000005e-25

   1. Initial program 65.5%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 21.4%

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

   4. Applied egg-rr 21.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{{k}^{2}}}{\color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({k}^{2}\right)\right), \left(e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      6. exp-diff N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right)\right) \]

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      18. *-lowering-*.f64 69.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 69.1%

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

      1. div-inv N/A

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

      2. times-frac N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot t}{\ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}\right)}\right) \]

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

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 93.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}} \]

   if 1.20000000000000005e-25 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 49.0%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. associate-*r* N/A

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

      10. associate-/r* N/A

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

   6. Applied egg-rr 51.8%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-/l* N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 54.9%

      \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \frac{1 + \left(t \cdot t\right) \cdot 0.3333333333333333}{\ell \cdot \ell}\right)}} \]

   10. Taylor expanded in l around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

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

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

   12. Simplified 61.2%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(2 \cdot \left(t \cdot t\right) + k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(2 \cdot \left(t \cdot t\right) + k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell} \cdot \frac{2 \cdot \left(t\_m \cdot t\_m\right) + k \cdot \left(k \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.1e-53)
    (/
     (/ 2.0 t_m)
     (*
      (/ (* k k) l)
      (/
       (+
        (* 2.0 (* t_m t_m))
        (* k (* k (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))))
       l)))
    (* (/ l k) (/ l (* t_m (* 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 <= 1.1e-53) {
		tmp = (2.0 / t_m) / (((k * k) / l) * (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) / l));
	} else {
		tmp = (l / k) * (l / (t_m * (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 <= 1.1d-53) then
        tmp = (2.0d0 / t_m) / (((k * k) / l) * (((2.0d0 * (t_m * t_m)) + (k * (k * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))))) / l))
    else
        tmp = (l / k) * (l / (t_m * (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 <= 1.1e-53) {
		tmp = (2.0 / t_m) / (((k * k) / l) * (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) / l));
	} else {
		tmp = (l / k) * (l / (t_m * (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 <= 1.1e-53:
		tmp = (2.0 / t_m) / (((k * k) / l) * (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) / l))
	else:
		tmp = (l / k) * (l / (t_m * (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 <= 1.1e-53)
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(k * k) / l) * Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(k * Float64(k * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))))) / l)));
	else
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(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 <= 1.1e-53)
		tmp = (2.0 / t_m) / (((k * k) / l) * (((2.0 * (t_m * t_m)) + (k * (k * (1.0 + ((t_m * t_m) * 0.3333333333333333))))) / l));
	else
		tmp = (l / k) * (l / (t_m * (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, 1.1e-53], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(k * N[(k * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.10000000000000009e-53

   1. Initial program 51.6%

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

      1. /-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 48.2%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. associate-*r* N/A

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

      10. associate-/r* N/A

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

   6. Applied egg-rr 50.2%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-/l* N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

   9. Simplified 54.2%

      \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \frac{1 + \left(t \cdot t\right) \cdot 0.3333333333333333}{\ell \cdot \ell}\right)}} \]

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right)}{{\ell}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

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

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

   12. Simplified 68.6%

       \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{2 \cdot \left(t \cdot t\right) + k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right)\right)}{\ell}}} \]

   if 1.10000000000000009e-53 < t

   1. Initial program 66.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.9%

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

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

      5. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 56.5%

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

   7. Simplified 56.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 76.6%

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

   11. Applied egg-rr 76.6%

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell} \cdot \frac{2 \cdot \left(t \cdot t\right) + k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.9% accurate, 15.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.38 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{1}{k \cdot k}}{t\_m \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{2 \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= k 1.3e-155)
    (* (/ l k) (/ l (* t_m (* t_m (* t_m k)))))
    (if (<= k 1.38e+54)
      (* (/ l t_m) (/ (/ 1.0 (* k k)) (* t_m (/ t_m l))))
      (/ (/ 2.0 t_m) (/ (* 2.0 (* t_m (* t_m (* k k)))) (* l 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 (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 1.38e+54) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (2.0 / t_m) / ((2.0 * (t_m * (t_m * (k * k)))) / (l * 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 (k <= 1.3d-155) then
        tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
    else if (k <= 1.38d+54) then
        tmp = (l / t_m) * ((1.0d0 / (k * k)) / (t_m * (t_m / l)))
    else
        tmp = (2.0d0 / t_m) / ((2.0d0 * (t_m * (t_m * (k * k)))) / (l * 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 (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else if (k <= 1.38e+54) {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	} else {
		tmp = (2.0 / t_m) / ((2.0 * (t_m * (t_m * (k * k)))) / (l * 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 k <= 1.3e-155:
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
	elif k <= 1.38e+54:
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)))
	else:
		tmp = (2.0 / t_m) / ((2.0 * (t_m * (t_m * (k * k)))) / (l * 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 (k <= 1.3e-155)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * k)))));
	elseif (k <= 1.38e+54)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(1.0 / Float64(k * k)) / Float64(t_m * Float64(t_m / l))));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(2.0 * Float64(t_m * Float64(t_m * Float64(k * k)))) / Float64(l * 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 (k <= 1.3e-155)
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	elseif (k <= 1.38e+54)
		tmp = (l / t_m) * ((1.0 / (k * k)) / (t_m * (t_m / l)));
	else
		tmp = (2.0 / t_m) / ((2.0 * (t_m * (t_m * (k * k)))) / (l * 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[k, 1.3e-155], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.38e+54], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(2.0 * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 50.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{\ell}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 67.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \left(\left(k \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot t\right), t\right), t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot k\right), t\right), t\right)\right)\right) \]

       6. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), t\right), t\right)\right)\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot t\right) \cdot t}} \]

   if 1.30000000000000004e-155 < k < 1.38e54

   1. Initial program 66.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{e^{\log t \cdot 3}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{e^{\log t \cdot 3}}{{\ell}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{e^{\log t \cdot 3}}{e^{\log \ell \cdot 2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. div-exp N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log t \cdot 3 - \log \ell \cdot 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log t \cdot 3\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(3 \cdot \log t\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \log t\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \mathsf{*.f64}\left(\log \ell, 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. log-lowering-log.f64 24.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 24.9%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{{k}^{2}}}{\color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{k}^{2}}\right), \color{blue}{\left(e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({k}^{2}\right)\right), \left(e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      6. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{e^{3 \cdot \log t}}{\color{blue}{e^{2 \cdot \log \ell}}}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{e^{\log t \cdot 3}}{e^{\color{blue}{2} \cdot \log \ell}}\right)\right) \]

      8. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{t}^{3}}{e^{\color{blue}{2 \cdot \log \ell}}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{t}^{3}}{e^{\log \ell \cdot 2}}\right)\right) \]

      10. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{t}^{3}}{{\ell}^{\color{blue}{2}}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left({t}^{3}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      18. *-lowering-*.f64 68.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}} \]

      2. times-frac N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot t}{\ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{t \cdot t}}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t \cdot \color{blue}{t}}{\ell}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 82.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}} \]

   if 1.38e54 < k

   1. Initial program 38.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-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 38.7%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\left(t \cdot t\right) \cdot \frac{1}{\color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}}}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot t}{\color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(t \cdot t\right), \color{blue}{\left(\frac{\ell \cdot \ell}{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{\ell \cdot \ell}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}{\color{blue}{2 + \frac{k \cdot \frac{k}{t}}{t}}}\right)\right)\right) \]

   6. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{t \cdot t}{\frac{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{2}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{2}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{\color{blue}{2}} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{\color{blue}{2}} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(\frac{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\frac{\color{blue}{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{{t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right)\right) \]

   9. Simplified 47.1%

      \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \frac{1 + \left(t \cdot t\right) \cdot 0.3333333333333333}{\ell \cdot \ell}\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}{\color{blue}{{\ell}^{2}}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{2 \cdot \left({t}^{2} \cdot {k}^{2}\right)}{{\ell}^{2}}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(2 \cdot {t}^{2}\right) \cdot {k}^{2}}{{\color{blue}{\ell}}^{2}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(2 \cdot {t}^{2}\right) \cdot {k}^{2}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(2 \cdot \left({t}^{2} \cdot {k}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot {t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({k}^{2} \cdot {t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({t}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(t \cdot t\right) \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(t \cdot {k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot {k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left({k}^{2}\right)\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

       16. *-lowering-*.f64 54.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   12. Simplified 54.9%

       \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}{\ell \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.38 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{2 \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{1}{k \cdot k}}{t\_m \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 1.3e-155)
    (* (/ l k) (/ l (* t_m (* t_m (* t_m k)))))
    (* (/ l t_m) (/ (/ 1.0 (* k k)) (* 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 (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (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 (k <= 1.3d-155) then
        tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
    else
        tmp = (l / t_m) * ((1.0d0 / (k * k)) / (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 (k <= 1.3e-155) {
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * ((1.0 / (k * k)) / (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 k <= 1.3e-155:
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))))
	else:
		tmp = (l / t_m) * ((1.0 / (k * k)) / (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 (k <= 1.3e-155)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(1.0 / Float64(k * k)) / 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)
	tmp = 0.0;
	if (k <= 1.3e-155)
		tmp = (l / k) * (l / (t_m * (t_m * (t_m * k))));
	else
		tmp = (l / t_m) * ((1.0 / (k * k)) / (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[k, 1.3e-155], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.30000000000000004e-155

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 50.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{\ell}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 67.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \left(\left(k \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot t\right), t\right), t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot k\right), t\right), t\right)\right)\right) \]

       6. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), t\right), t\right)\right)\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot t\right) \cdot t}} \]

   if 1.30000000000000004e-155 < k

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{e^{\log t \cdot 3}}{\ell \cdot \ell}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{e^{\log t \cdot 3}}{{\ell}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{e^{\log t \cdot 3}}{e^{\log \ell \cdot 2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. div-exp N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log t \cdot 3 - \log \ell \cdot 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log t \cdot 3\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(3 \cdot \log t\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \log t\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \left(\log \ell \cdot 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \mathsf{*.f64}\left(\log \ell, 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. log-lowering-log.f64 20.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 20.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{{k}^{2}}}{\color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{k}^{2}}\right), \color{blue}{\left(e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({k}^{2}\right)\right), \left(e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(e^{3 \cdot \log t - \color{blue}{2 \cdot \log \ell}}\right)\right) \]

      6. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{e^{3 \cdot \log t}}{\color{blue}{e^{2 \cdot \log \ell}}}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{e^{\log t \cdot 3}}{e^{\color{blue}{2} \cdot \log \ell}}\right)\right) \]

      8. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{t}^{3}}{e^{\color{blue}{2 \cdot \log \ell}}}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{t}^{3}}{e^{\log \ell \cdot 2}}\right)\right) \]

      10. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{t}^{3}}{{\ell}^{\color{blue}{2}}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left({t}^{3}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(t \cdot {t}^{2}\right), \left({\ell}^{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      18. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}} \]

      2. times-frac N/A

         \[\leadsto \frac{1 \cdot \frac{1}{k \cdot k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot t}{\ell}}} \]

      3. times-frac N/A

         \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{1}{k \cdot k}}{\frac{t \cdot t}{\ell}}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\frac{1}{k \cdot k}}}{\frac{t \cdot t}{\ell}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{1}{k \cdot k}\right), \color{blue}{\left(\frac{t \cdot t}{\ell}\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{t \cdot t}}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t \cdot \color{blue}{t}}{\ell}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 67.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\frac{\frac{\frac{\ell}{k}}{t\_m}}{t\_m}}{t\_m}}{k}\right) \end{array} \]

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 (/ (/ (/ (/ (/ l k) t_m) 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 * (((((l / k) / t_m) / 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 * (((((l / k) / t_m) / 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 * (((((l / k) / t_m) / 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 * (((((l / k) / t_m) / 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(l * Float64(Float64(Float64(Float64(Float64(l / k) / t_m) / 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 * (((((l / k) / t_m) / 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[(l * N[(N[(N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 51.3%

   \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 51.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

7. Simplified 51.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{\ell}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 63.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 63.7%

   \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{\ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{\color{blue}{k}} \]

    2. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{k}\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}\right), k\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \left(t \cdot \left(t \cdot t\right)\right)\right), k\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(t \cdot \left(t \cdot t\right)\right)\right), k\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), k\right)\right) \]

    9. *-lowering-*.f64 62.3%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), k\right)\right) \]

11. Applied egg-rr 62.3%

    \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}}{k}} \]
12. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{k}}{t}}{t \cdot t}\right), k\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\frac{\ell}{k}}{t}}{t}}{t}\right), k\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{k}}{t}}{t}\right), t\right), k\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t}\right), t\right), t\right), k\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), t\right), t\right), t\right), k\right)\right) \]

    6. /-lowering-/.f64 69.5%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), t\right), t\right), t\right), k\right)\right) \]

13. Applied egg-rr 69.5%

    \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\frac{\frac{\ell}{k}}{t}}{t}}{t}}}{k} \]
14. Add Preprocessing

Alternative 12: 63.9% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\frac{\frac{\ell}{k}}{t\_m \cdot t\_m}}{t\_m}}{k}\right) \end{array} \]

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 (/ (/ (/ (/ l k) (* t_m 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 * ((((l / k) / (t_m * 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 * ((((l / k) / (t_m * 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 * ((((l / k) / (t_m * 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 * ((((l / k) / (t_m * 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(l * Float64(Float64(Float64(Float64(l / k) / Float64(t_m * 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 * ((((l / k) / (t_m * 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[(l * N[(N[(N[(N[(l / k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 51.3%

   \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 51.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

7. Simplified 51.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{\ell}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 63.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 63.7%

   \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{\ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{\color{blue}{k}} \]

    2. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{k}\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}\right), k\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \left(t \cdot \left(t \cdot t\right)\right)\right), k\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(t \cdot \left(t \cdot t\right)\right)\right), k\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), k\right)\right) \]

    9. *-lowering-*.f64 62.3%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), k\right)\right) \]

11. Applied egg-rr 62.3%

    \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}}{k}} \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{\left(t \cdot t\right) \cdot t}\right), k\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t}\right), k\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t \cdot t}\right), t\right), k\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \left(t \cdot t\right)\right), t\right), k\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(t \cdot t\right)\right), t\right), k\right)\right) \]

    6. *-lowering-*.f64 64.1%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, t\right)\right), t\right), k\right)\right) \]

13. Applied egg-rr 64.1%

    \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t}}}{k} \]
14. Add Preprocessing

Alternative 13: 63.3% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}}{k}\right) \end{array} \]

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 (/ (/ l (* t_m (* k (* 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 * ((l / (t_m * (k * (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 * ((l / (t_m * (k * (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 * ((l / (t_m * (k * (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 * ((l / (t_m * (k * (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(l * Float64(Float64(l / Float64(t_m * Float64(k * 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 * ((l / (t_m * (k * (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[(l * N[(N[(l / N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 51.3%

   \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 51.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

7. Simplified 51.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{\ell}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 63.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 63.7%

   \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{\ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{\color{blue}{k}} \]

    2. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{k}\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}\right), k\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \left(t \cdot \left(t \cdot t\right)\right)\right), k\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(t \cdot \left(t \cdot t\right)\right)\right), k\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right), k\right)\right) \]

    9. *-lowering-*.f64 62.3%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right), k\right)\right) \]

11. Applied egg-rr 62.3%

    \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}}{k}} \]
12. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k}\right), k\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), k\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right), k\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(t \cdot t\right) \cdot k\right)\right)\right), k\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), k\right)\right) \]

    6. *-lowering-*.f64 64.0%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), k\right)\right) \]

13. Applied egg-rr 64.0%

    \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{k} \]

14. Final simplification 64.0%

    \[\leadsto \ell \cdot \frac{\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}}{k} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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))))