Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 83.8%

Time: 18.2s

Alternatives: 21

Speedup: 32.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{l\_m}{{t\_m}^{1.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{l\_m \cdot l\_m}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{t\_2} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot e^{3 \cdot \log t\_m - 2 \cdot \log l\_m}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ l_m (pow t_m 1.5))))
   (*
    t_s
    (if (<= t_m 5.5e-168)
      (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l_m l_m)) (cos k)))
      (if (<= t_m 1.9e+204)
        (/
         2.0
         (/
          (* (/ (sin k) t_2) (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))))
          t_2))
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (exp (- (* 3.0 (log t_m)) (* 2.0 (log l_m))))))
          (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = l_m / pow(t_m, 1.5);
	double tmp;
	if (t_m <= 5.5e-168) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l_m * l_m)) / cos(k));
	} else if (t_m <= 1.9e+204) {
		tmp = 2.0 / (((sin(k) / t_2) * (tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m)))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l_m / (t_m ** 1.5d0)
    if (t_m <= 5.5d-168) then
        tmp = 2.0d0 / (((k * (k * (t_m * (sin(k) ** 2.0d0)))) / (l_m * l_m)) / cos(k))
    else if (t_m <= 1.9d+204) then
        tmp = 2.0d0 / (((sin(k) / t_2) * (tan(k) * (2.0d0 + ((k / t_m) / (t_m / k))))) / t_2)
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * exp(((3.0d0 * log(t_m)) - (2.0d0 * log(l_m)))))) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = l_m / Math.pow(t_m, 1.5);
	double tmp;
	if (t_m <= 5.5e-168) {
		tmp = 2.0 / (((k * (k * (t_m * Math.pow(Math.sin(k), 2.0)))) / (l_m * l_m)) / Math.cos(k));
	} else if (t_m <= 1.9e+204) {
		tmp = 2.0 / (((Math.sin(k) / t_2) * (Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.exp(((3.0 * Math.log(t_m)) - (2.0 * Math.log(l_m)))))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = l_m / math.pow(t_m, 1.5)
	tmp = 0
	if t_m <= 5.5e-168:
		tmp = 2.0 / (((k * (k * (t_m * math.pow(math.sin(k), 2.0)))) / (l_m * l_m)) / math.cos(k))
	elif t_m <= 1.9e+204:
		tmp = 2.0 / (((math.sin(k) / t_2) * (math.tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2)
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.exp(((3.0 * math.log(t_m)) - (2.0 * math.log(l_m)))))) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(l_m / (t_m ^ 1.5))
	tmp = 0.0
	if (t_m <= 5.5e-168)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l_m * l_m)) / cos(k)));
	elseif (t_m <= 1.9e+204)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / t_2) * Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))) / t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * exp(Float64(Float64(3.0 * log(t_m)) - Float64(2.0 * log(l_m)))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = l_m / (t_m ^ 1.5);
	tmp = 0.0;
	if (t_m <= 5.5e-168)
		tmp = 2.0 / (((k * (k * (t_m * (sin(k) ^ 2.0)))) / (l_m * l_m)) / cos(k));
	elseif (t_m <= 1.9e+204)
		tmp = 2.0 / (((sin(k) / t_2) * (tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m)))))) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(l$95$m / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.5e-168], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+204], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $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 < 5.4999999999999999e-168

   1. Initial program 55.1%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 11.0%

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

   4. Applied egg-rr 11.0%

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

   5. Taylor expanded in k around inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. cos-lowering-cos.f64 69.8%

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

   7. Simplified 69.8%

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

   if 5.4999999999999999e-168 < t < 1.8999999999999999e204

   1. Initial program 64.4%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. div-inv N/A

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

      2. un-div-inv N/A

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

      3. associate-*l* N/A

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

      4. un-div-inv N/A

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

      5. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 91.7%

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

   if 1.8999999999999999e204 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.3%

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

   4. Applied egg-rr 47.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{\frac{\ell}{{t}^{1.5}}} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}{\frac{\ell}{{t}^{1.5}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{l\_m}{{t\_m}^{1.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{l\_m \cdot l\_m}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{t\_2} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot e^{3 \cdot \log t\_m - 2 \cdot \log l\_m}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ l_m (pow t_m 1.5))))
   (*
    t_s
    (if (<= t_m 5.5e-168)
      (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l_m l_m)) (cos k)))
      (if (<= t_m 3.2e+205)
        (/
         2.0
         (/
          (* (/ (sin k) t_2) (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))))
          t_2))
        (/
         2.0
         (*
          2.0
          (*
           (tan k)
           (* (sin k) (exp (- (* 3.0 (log t_m)) (* 2.0 (log l_m)))))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = l_m / pow(t_m, 1.5);
	double tmp;
	if (t_m <= 5.5e-168) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l_m * l_m)) / cos(k));
	} else if (t_m <= 3.2e+205) {
		tmp = 2.0 / (((sin(k) / t_2) * (tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m)))))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l_m / (t_m ** 1.5d0)
    if (t_m <= 5.5d-168) then
        tmp = 2.0d0 / (((k * (k * (t_m * (sin(k) ** 2.0d0)))) / (l_m * l_m)) / cos(k))
    else if (t_m <= 3.2d+205) then
        tmp = 2.0d0 / (((sin(k) / t_2) * (tan(k) * (2.0d0 + ((k / t_m) / (t_m / k))))) / t_2)
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (sin(k) * exp(((3.0d0 * log(t_m)) - (2.0d0 * log(l_m)))))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = l_m / Math.pow(t_m, 1.5);
	double tmp;
	if (t_m <= 5.5e-168) {
		tmp = 2.0 / (((k * (k * (t_m * Math.pow(Math.sin(k), 2.0)))) / (l_m * l_m)) / Math.cos(k));
	} else if (t_m <= 3.2e+205) {
		tmp = 2.0 / (((Math.sin(k) / t_2) * (Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (Math.sin(k) * Math.exp(((3.0 * Math.log(t_m)) - (2.0 * Math.log(l_m)))))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = l_m / math.pow(t_m, 1.5)
	tmp = 0
	if t_m <= 5.5e-168:
		tmp = 2.0 / (((k * (k * (t_m * math.pow(math.sin(k), 2.0)))) / (l_m * l_m)) / math.cos(k))
	elif t_m <= 3.2e+205:
		tmp = 2.0 / (((math.sin(k) / t_2) * (math.tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2)
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * (math.sin(k) * math.exp(((3.0 * math.log(t_m)) - (2.0 * math.log(l_m)))))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(l_m / (t_m ^ 1.5))
	tmp = 0.0
	if (t_m <= 5.5e-168)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l_m * l_m)) / cos(k)));
	elseif (t_m <= 3.2e+205)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / t_2) * Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))) / t_2));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(sin(k) * exp(Float64(Float64(3.0 * log(t_m)) - Float64(2.0 * log(l_m))))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = l_m / (t_m ^ 1.5);
	tmp = 0.0;
	if (t_m <= 5.5e-168)
		tmp = 2.0 / (((k * (k * (t_m * (sin(k) ^ 2.0)))) / (l_m * l_m)) / cos(k));
	elseif (t_m <= 3.2e+205)
		tmp = 2.0 / (((sin(k) / t_2) * (tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	else
		tmp = 2.0 / (2.0 * (tan(k) * (sin(k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m)))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(l$95$m / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.5e-168], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+205], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.4999999999999999e-168

   1. Initial program 55.1%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 11.0%

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

   4. Applied egg-rr 11.0%

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

   5. Taylor expanded in k around inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. cos-lowering-cos.f64 69.8%

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

   7. Simplified 69.8%

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

   if 5.4999999999999999e-168 < t < 3.19999999999999996e205

   1. Initial program 64.4%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. div-inv N/A

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

      2. un-div-inv N/A

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

      3. associate-*l* N/A

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

      4. un-div-inv N/A

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

      5. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 91.7%

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

   if 3.19999999999999996e205 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 47.3%

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), 2\right)\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \color{blue}{2}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 47.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{\frac{\ell}{{t}^{1.5}}} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}{\frac{\ell}{{t}^{1.5}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{l\_m}{{t\_m}^{1.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{l\_m \cdot l\_m}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{t\_2} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{3 \cdot \log t\_m + 2 \cdot \log k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ l_m (pow t_m 1.5))))
   (*
    t_s
    (if (<= t_m 9.5e-168)
      (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l_m l_m)) (cos k)))
      (if (<= t_m 1.9e+204)
        (/
         2.0
         (/
          (* (/ (sin k) t_2) (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))))
          t_2))
        (* l_m (/ l_m (exp (+ (* 3.0 (log t_m)) (* 2.0 (log k)))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = l_m / pow(t_m, 1.5);
	double tmp;
	if (t_m <= 9.5e-168) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l_m * l_m)) / cos(k));
	} else if (t_m <= 1.9e+204) {
		tmp = 2.0 / (((sin(k) / t_2) * (tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	} else {
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l_m / (t_m ** 1.5d0)
    if (t_m <= 9.5d-168) then
        tmp = 2.0d0 / (((k * (k * (t_m * (sin(k) ** 2.0d0)))) / (l_m * l_m)) / cos(k))
    else if (t_m <= 1.9d+204) then
        tmp = 2.0d0 / (((sin(k) / t_2) * (tan(k) * (2.0d0 + ((k / t_m) / (t_m / k))))) / t_2)
    else
        tmp = l_m * (l_m / exp(((3.0d0 * log(t_m)) + (2.0d0 * log(k)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = l_m / Math.pow(t_m, 1.5);
	double tmp;
	if (t_m <= 9.5e-168) {
		tmp = 2.0 / (((k * (k * (t_m * Math.pow(Math.sin(k), 2.0)))) / (l_m * l_m)) / Math.cos(k));
	} else if (t_m <= 1.9e+204) {
		tmp = 2.0 / (((Math.sin(k) / t_2) * (Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	} else {
		tmp = l_m * (l_m / Math.exp(((3.0 * Math.log(t_m)) + (2.0 * Math.log(k)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = l_m / math.pow(t_m, 1.5)
	tmp = 0
	if t_m <= 9.5e-168:
		tmp = 2.0 / (((k * (k * (t_m * math.pow(math.sin(k), 2.0)))) / (l_m * l_m)) / math.cos(k))
	elif t_m <= 1.9e+204:
		tmp = 2.0 / (((math.sin(k) / t_2) * (math.tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2)
	else:
		tmp = l_m * (l_m / math.exp(((3.0 * math.log(t_m)) + (2.0 * math.log(k)))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(l_m / (t_m ^ 1.5))
	tmp = 0.0
	if (t_m <= 9.5e-168)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l_m * l_m)) / cos(k)));
	elseif (t_m <= 1.9e+204)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / t_2) * Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))) / t_2));
	else
		tmp = Float64(l_m * Float64(l_m / exp(Float64(Float64(3.0 * log(t_m)) + Float64(2.0 * log(k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = l_m / (t_m ^ 1.5);
	tmp = 0.0;
	if (t_m <= 9.5e-168)
		tmp = 2.0 / (((k * (k * (t_m * (sin(k) ^ 2.0)))) / (l_m * l_m)) / cos(k));
	elseif (t_m <= 1.9e+204)
		tmp = 2.0 / (((sin(k) / t_2) * (tan(k) * (2.0 + ((k / t_m) / (t_m / k))))) / t_2);
	else
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(l$95$m / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-168], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+204], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 9.49999999999999918e-168

   1. Initial program 55.1%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 11.0%

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

   4. Applied egg-rr 11.0%

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

   5. Taylor expanded in k around inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. cos-lowering-cos.f64 69.8%

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

   7. Simplified 69.8%

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

   if 9.49999999999999918e-168 < t < 1.8999999999999999e204

   1. Initial program 64.4%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. div-inv N/A

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

      2. un-div-inv N/A

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

      3. associate-*l* N/A

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

      4. un-div-inv N/A

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

      5. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 91.7%

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

   if 1.8999999999999999e204 < t

   1. Initial program 71.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 71.9%

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

   5. Simplified 71.9%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 57.5%

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

   7. Applied egg-rr 57.5%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 58.0%

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

   9. Applied egg-rr 58.0%

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

       1. cube-unmult N/A

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

       2. pow-to-exp N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. pow-to-exp N/A

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

       6. prod-exp N/A

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

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

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

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

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

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

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

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

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

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

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

       12. log-lowering-log.f64 37.6%

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

   11. Applied egg-rr 37.6%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{e^{\log k \cdot 2 + 3 \cdot \log t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k}{\frac{\ell}{{t}^{1.5}}} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}{\frac{\ell}{{t}^{1.5}}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{e^{3 \cdot \log t + 2 \cdot \log k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{l\_m \cdot l\_m}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\left(\sin k \cdot \frac{t\_m \cdot t\_m}{l\_m}\right) \cdot \frac{\tan k}{\frac{l\_m}{t\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t\_m \cdot t\_m}\right)\right)} \cdot \frac{2}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-134)
    (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l_m l_m)) (cos k)))
    (if (<= t_m 1.05e+124)
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (* (sin k) (/ (* t_m t_m) l_m)) (/ (tan k) (/ l_m t_m)))))
      (/
       (*
        (/
         (* (/ l_m t_m) (/ l_m k))
         (+ 2.0 (* k (* k (+ 0.3333333333333333 (/ 1.0 (* t_m t_m)))))))
        (/ 2.0 t_m))
       (* t_m k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.8e-134) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l_m * l_m)) / cos(k));
	} else if (t_m <= 1.05e+124) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * ((sin(k) * ((t_m * t_m) / l_m)) * (tan(k) / (l_m / t_m))));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.8d-134) then
        tmp = 2.0d0 / (((k * (k * (t_m * (sin(k) ** 2.0d0)))) / (l_m * l_m)) / cos(k))
    else if (t_m <= 1.05d+124) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * ((sin(k) * ((t_m * t_m) / l_m)) * (tan(k) / (l_m / t_m))))
    else
        tmp = ((((l_m / t_m) * (l_m / k)) / (2.0d0 + (k * (k * (0.3333333333333333d0 + (1.0d0 / (t_m * t_m))))))) * (2.0d0 / t_m)) / (t_m * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.8e-134) {
		tmp = 2.0 / (((k * (k * (t_m * Math.pow(Math.sin(k), 2.0)))) / (l_m * l_m)) / Math.cos(k));
	} else if (t_m <= 1.05e+124) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * ((Math.sin(k) * ((t_m * t_m) / l_m)) * (Math.tan(k) / (l_m / t_m))));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 2.8e-134:
		tmp = 2.0 / (((k * (k * (t_m * math.pow(math.sin(k), 2.0)))) / (l_m * l_m)) / math.cos(k))
	elif t_m <= 1.05e+124:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * ((math.sin(k) * ((t_m * t_m) / l_m)) * (math.tan(k) / (l_m / t_m))))
	else:
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.8e-134)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l_m * l_m)) / cos(k)));
	elseif (t_m <= 1.05e+124)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(sin(k) * Float64(Float64(t_m * t_m) / l_m)) * Float64(tan(k) / Float64(l_m / t_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(2.0 + Float64(k * Float64(k * Float64(0.3333333333333333 + Float64(1.0 / Float64(t_m * t_m))))))) * Float64(2.0 / t_m)) / Float64(t_m * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 2.8e-134)
		tmp = 2.0 / (((k * (k * (t_m * (sin(k) ^ 2.0)))) / (l_m * l_m)) / cos(k));
	elseif (t_m <= 1.05e+124)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * ((sin(k) * ((t_m * t_m) / l_m)) * (tan(k) / (l_m / t_m))));
	else
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-134], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+124], 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[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(k * N[(k * N[(0.3333333333333333 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.7999999999999999e-134

   1. Initial program 54.4%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 12.5%

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

   4. Applied egg-rr 12.5%

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

   5. Taylor expanded in k around inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. cos-lowering-cos.f64 69.9%

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

   7. Simplified 69.9%

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

   if 2.7999999999999999e-134 < t < 1.05000000000000006e124

   1. Initial program 70.8%

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

      1. 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 42.0%

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

   4. Applied egg-rr 42.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. div-exp N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. cube-unmult N/A

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

      8. pow-to-exp N/A

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

      9. pow2 N/A

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

      10. clear-num N/A

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

      11. associate-*r* N/A

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

      12. frac-times N/A

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

      13. div-inv N/A

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

      14. frac-times N/A

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

      15. clear-num N/A

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

      16. associate-/l* N/A

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

      17. associate-*r* N/A

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

   6. Applied egg-rr 91.8%

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

   if 1.05000000000000006e124 < t

   1. Initial program 63.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 59.4%

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

   7. Simplified 59.4%

      \[\leadsto \frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}} \]
   8. Step-by-step derivation

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 72.4%

      \[\leadsto \frac{2}{t \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}}{k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. frac-times N/A

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

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

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

   11. Applied egg-rr 82.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{k \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\sin k \cdot \frac{t \cdot t}{\ell}\right) \cdot \frac{\tan k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{l\_m \cdot l\_m}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+126}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{\sin k \cdot \left(t\_m \cdot t\_m\right)}{l\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t\_m \cdot t\_m}\right)\right)} \cdot \frac{2}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-42)
    (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l_m l_m)) (cos k)))
    (if (<= t_m 5.6e+126)
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (tan k) (* (/ t_m l_m) (/ (* (sin k) (* t_m t_m)) l_m)))))
      (/
       (*
        (/
         (* (/ l_m t_m) (/ l_m k))
         (+ 2.0 (* k (* k (+ 0.3333333333333333 (/ 1.0 (* t_m t_m)))))))
        (/ 2.0 t_m))
       (* t_m k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.2e-42) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l_m * l_m)) / cos(k));
	} else if (t_m <= 5.6e+126) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * ((t_m / l_m) * ((sin(k) * (t_m * t_m)) / l_m))));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.2d-42) then
        tmp = 2.0d0 / (((k * (k * (t_m * (sin(k) ** 2.0d0)))) / (l_m * l_m)) / cos(k))
    else if (t_m <= 5.6d+126) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (tan(k) * ((t_m / l_m) * ((sin(k) * (t_m * t_m)) / l_m))))
    else
        tmp = ((((l_m / t_m) * (l_m / k)) / (2.0d0 + (k * (k * (0.3333333333333333d0 + (1.0d0 / (t_m * t_m))))))) * (2.0d0 / t_m)) / (t_m * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.2e-42) {
		tmp = 2.0 / (((k * (k * (t_m * Math.pow(Math.sin(k), 2.0)))) / (l_m * l_m)) / Math.cos(k));
	} else if (t_m <= 5.6e+126) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * ((t_m / l_m) * ((Math.sin(k) * (t_m * t_m)) / l_m))));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.2e-42:
		tmp = 2.0 / (((k * (k * (t_m * math.pow(math.sin(k), 2.0)))) / (l_m * l_m)) / math.cos(k))
	elif t_m <= 5.6e+126:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (math.tan(k) * ((t_m / l_m) * ((math.sin(k) * (t_m * t_m)) / l_m))))
	else:
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.2e-42)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l_m * l_m)) / cos(k)));
	elseif (t_m <= 5.6e+126)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(Float64(t_m / l_m) * Float64(Float64(sin(k) * Float64(t_m * t_m)) / l_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(2.0 + Float64(k * Float64(k * Float64(0.3333333333333333 + Float64(1.0 / Float64(t_m * t_m))))))) * Float64(2.0 / t_m)) / Float64(t_m * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.2e-42)
		tmp = 2.0 / (((k * (k * (t_m * (sin(k) ^ 2.0)))) / (l_m * l_m)) / cos(k));
	elseif (t_m <= 5.6e+126)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (tan(k) * ((t_m / l_m) * ((sin(k) * (t_m * t_m)) / l_m))));
	else
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-42], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+126], 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[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(k * N[(k * N[(0.3333333333333333 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.2000000000000004e-42

   1. Initial program 55.6%

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

      1. *-commutative N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

      5. cube-unmult N/A

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

      6. sqr-pow N/A

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

      7. times-frac N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. metadata-eval 20.3%

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

   4. Applied egg-rr 20.3%

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

   5. Taylor expanded in k around inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. cos-lowering-cos.f64 71.1%

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

   7. Simplified 71.1%

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

   if 7.2000000000000004e-42 < t < 5.60000000000000018e126

   1. Initial program 73.5%

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

      1. cube-unmult N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      10. sin-lowering-sin.f64 92.6%

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

   4. Applied egg-rr 92.6%

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

   if 5.60000000000000018e126 < t

   1. Initial program 63.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 59.4%

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

   7. Simplified 59.4%

      \[\leadsto \frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}} \]
   8. Step-by-step derivation

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 72.4%

      \[\leadsto \frac{2}{t \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}}{k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. frac-times N/A

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

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

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

   11. Applied egg-rr 82.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{k \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+126}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k \cdot \left(t \cdot t\right)}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-160}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{3 \cdot \log t\_m + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{l\_m}{t\_m} \cdot {\left(t\_m \cdot \frac{\sin k \cdot \tan k}{\frac{l\_m}{t\_m}}\right)}^{-1}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}{\left(l\_m \cdot l\_m\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m}}{2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}} \cdot \left(\frac{2}{t\_m} \cdot \frac{\frac{l\_m}{\tan k}}{\sin k}\right)}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.75e-160)
    (* l_m (/ l_m (exp (+ (* 3.0 (log t_m)) (* 2.0 (log k))))))
    (if (<= k 1.7e-12)
      (* (/ l_m t_m) (pow (* t_m (/ (* (sin k) (tan k)) (/ l_m t_m))) -1.0))
      (if (<= k 2e+83)
        (/
         2.0
         (/ (* (pow (sin k) 2.0) (* t_m (* k k))) (* (* l_m l_m) (cos k))))
        (/
         (*
          (/ (/ l_m t_m) (+ 2.0 (/ k (/ (* t_m t_m) k))))
          (* (/ 2.0 t_m) (/ (/ l_m (tan k)) (sin k))))
         t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.75e-160) {
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	} else if (k <= 1.7e-12) {
		tmp = (l_m / t_m) * pow((t_m * ((sin(k) * tan(k)) / (l_m / t_m))), -1.0);
	} else if (k <= 2e+83) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t_m * (k * k))) / ((l_m * l_m) * cos(k)));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d-160) then
        tmp = l_m * (l_m / exp(((3.0d0 * log(t_m)) + (2.0d0 * log(k)))))
    else if (k <= 1.7d-12) then
        tmp = (l_m / t_m) * ((t_m * ((sin(k) * tan(k)) / (l_m / t_m))) ** (-1.0d0))
    else if (k <= 2d+83) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * (t_m * (k * k))) / ((l_m * l_m) * cos(k)))
    else
        tmp = (((l_m / t_m) / (2.0d0 + (k / ((t_m * t_m) / k)))) * ((2.0d0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.75e-160) {
		tmp = l_m * (l_m / Math.exp(((3.0 * Math.log(t_m)) + (2.0 * Math.log(k)))));
	} else if (k <= 1.7e-12) {
		tmp = (l_m / t_m) * Math.pow((t_m * ((Math.sin(k) * Math.tan(k)) / (l_m / t_m))), -1.0);
	} else if (k <= 2e+83) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t_m * (k * k))) / ((l_m * l_m) * Math.cos(k)));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / Math.tan(k)) / Math.sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.75e-160:
		tmp = l_m * (l_m / math.exp(((3.0 * math.log(t_m)) + (2.0 * math.log(k)))))
	elif k <= 1.7e-12:
		tmp = (l_m / t_m) * math.pow((t_m * ((math.sin(k) * math.tan(k)) / (l_m / t_m))), -1.0)
	elif k <= 2e+83:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * (t_m * (k * k))) / ((l_m * l_m) * math.cos(k)))
	else:
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / math.tan(k)) / math.sin(k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.75e-160)
		tmp = Float64(l_m * Float64(l_m / exp(Float64(Float64(3.0 * log(t_m)) + Float64(2.0 * log(k))))));
	elseif (k <= 1.7e-12)
		tmp = Float64(Float64(l_m / t_m) * (Float64(t_m * Float64(Float64(sin(k) * tan(k)) / Float64(l_m / t_m))) ^ -1.0));
	elseif (k <= 2e+83)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k))) / Float64(Float64(l_m * l_m) * cos(k))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / t_m) / Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) * Float64(Float64(2.0 / t_m) * Float64(Float64(l_m / tan(k)) / sin(k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.75e-160)
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	elseif (k <= 1.7e-12)
		tmp = (l_m / t_m) * ((t_m * ((sin(k) * tan(k)) / (l_m / t_m))) ^ -1.0);
	elseif (k <= 2e+83)
		tmp = 2.0 / (((sin(k) ^ 2.0) * (t_m * (k * k))) / ((l_m * l_m) * cos(k)));
	else
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.75e-160], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e-12], N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Power[N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e+83], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 1.7500000000000001e-160

   1. Initial program 64.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. cube-unmult N/A

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

       2. pow-to-exp N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. pow-to-exp N/A

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

       6. prod-exp N/A

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

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

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

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

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

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

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

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

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

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

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

       12. log-lowering-log.f64 10.7%

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

   11. Applied egg-rr 10.7%

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

   if 1.7500000000000001e-160 < k < 1.7e-12

   1. Initial program 53.8%

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

   3. Taylor expanded in t around inf

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 64.0%

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

   5. Simplified 64.0%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. cube-unmult N/A

         \[\leadsto \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      6. pow-to-exp N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. pow-to-exp N/A

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

      10. div-exp N/A

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

      11. unpow2 N/A

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

      12. associate-*r/ N/A

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

      13. tan-quot N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

   7. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

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

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

   9. Applied egg-rr 85.8%

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

   if 1.7e-12 < k < 2.00000000000000006e83

   1. Initial program 43.5%

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

      1. associate-*l* N/A

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

      2. cube-unmult N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      16. /-lowering-/.f64 52.5%

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

   4. Applied egg-rr 52.5%

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

   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. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 87.3%

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

   7. Simplified 87.3%

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

   if 2.00000000000000006e83 < k

   1. Initial program 51.2%

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

   4. Applied egg-rr 53.9%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

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

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

   6. Applied egg-rr 77.8%

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

      1. times-frac N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      7. associate-/l/ N/A

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

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

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

      9. associate-*l/ N/A

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

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

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

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

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

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

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

   8. Applied egg-rr 75.7%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{e^{3 \cdot \log t + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\ell}{t} \cdot {\left(t \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right)}^{-1}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{3 \cdot \log t\_m + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{l\_m}{t\_m} \cdot {\left(t\_m \cdot \frac{\sin k \cdot \tan k}{\frac{l\_m}{t\_m}}\right)}^{-1}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{+78}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{l\_m \cdot l\_m}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m}}{2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}} \cdot \left(\frac{2}{t\_m} \cdot \frac{\frac{l\_m}{\tan k}}{\sin k}\right)}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.8e-160)
    (* l_m (/ l_m (exp (+ (* 3.0 (log t_m)) (* 2.0 (log k))))))
    (if (<= k 1.5e-12)
      (* (/ l_m t_m) (pow (* t_m (/ (* (sin k) (tan k)) (/ l_m t_m))) -1.0))
      (if (<= k 1.85e+78)
        (*
         (/ 2.0 (* k k))
         (* (/ (* l_m l_m) t_m) (/ (cos k) (pow (sin k) 2.0))))
        (/
         (*
          (/ (/ l_m t_m) (+ 2.0 (/ k (/ (* t_m t_m) k))))
          (* (/ 2.0 t_m) (/ (/ l_m (tan k)) (sin k))))
         t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.8e-160) {
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	} else if (k <= 1.5e-12) {
		tmp = (l_m / t_m) * pow((t_m * ((sin(k) * tan(k)) / (l_m / t_m))), -1.0);
	} else if (k <= 1.85e+78) {
		tmp = (2.0 / (k * k)) * (((l_m * l_m) / t_m) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.8d-160) then
        tmp = l_m * (l_m / exp(((3.0d0 * log(t_m)) + (2.0d0 * log(k)))))
    else if (k <= 1.5d-12) then
        tmp = (l_m / t_m) * ((t_m * ((sin(k) * tan(k)) / (l_m / t_m))) ** (-1.0d0))
    else if (k <= 1.85d+78) then
        tmp = (2.0d0 / (k * k)) * (((l_m * l_m) / t_m) * (cos(k) / (sin(k) ** 2.0d0)))
    else
        tmp = (((l_m / t_m) / (2.0d0 + (k / ((t_m * t_m) / k)))) * ((2.0d0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.8e-160) {
		tmp = l_m * (l_m / Math.exp(((3.0 * Math.log(t_m)) + (2.0 * Math.log(k)))));
	} else if (k <= 1.5e-12) {
		tmp = (l_m / t_m) * Math.pow((t_m * ((Math.sin(k) * Math.tan(k)) / (l_m / t_m))), -1.0);
	} else if (k <= 1.85e+78) {
		tmp = (2.0 / (k * k)) * (((l_m * l_m) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / Math.tan(k)) / Math.sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.8e-160:
		tmp = l_m * (l_m / math.exp(((3.0 * math.log(t_m)) + (2.0 * math.log(k)))))
	elif k <= 1.5e-12:
		tmp = (l_m / t_m) * math.pow((t_m * ((math.sin(k) * math.tan(k)) / (l_m / t_m))), -1.0)
	elif k <= 1.85e+78:
		tmp = (2.0 / (k * k)) * (((l_m * l_m) / t_m) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / math.tan(k)) / math.sin(k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.8e-160)
		tmp = Float64(l_m * Float64(l_m / exp(Float64(Float64(3.0 * log(t_m)) + Float64(2.0 * log(k))))));
	elseif (k <= 1.5e-12)
		tmp = Float64(Float64(l_m / t_m) * (Float64(t_m * Float64(Float64(sin(k) * tan(k)) / Float64(l_m / t_m))) ^ -1.0));
	elseif (k <= 1.85e+78)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(Float64(l_m * l_m) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / t_m) / Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) * Float64(Float64(2.0 / t_m) * Float64(Float64(l_m / tan(k)) / sin(k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.8e-160)
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	elseif (k <= 1.5e-12)
		tmp = (l_m / t_m) * ((t_m * ((sin(k) * tan(k)) / (l_m / t_m))) ^ -1.0);
	elseif (k <= 1.85e+78)
		tmp = (2.0 / (k * k)) * (((l_m * l_m) / t_m) * (cos(k) / (sin(k) ^ 2.0)));
	else
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.8e-160], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e-12], N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Power[N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.85e+78], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 1.7999999999999999e-160

   1. Initial program 64.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 65.3%

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

   5. Simplified 65.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. frac-times N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. cube-unmult N/A

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

       2. pow-to-exp N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. pow-to-exp N/A

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

       6. prod-exp N/A

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

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

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

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

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

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

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

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

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

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(3, \log t\right)\right)\right)\right)\right) \]

       12. log-lowering-log.f64 10.7%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 10.7%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{e^{\log k \cdot 2 + 3 \cdot \log t}}} \]

   if 1.7999999999999999e-160 < k < 1.5000000000000001e-12

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \cos k\right), \color{blue}{\left({t}^{3} \cdot {\sin k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \cos k\right), \left({\color{blue}{t}}^{3} \cdot {\sin k}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \cos k\right)\right), \left({t}^{\color{blue}{3}} \cdot {\sin k}^{2}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({t}^{3} \cdot {\sin k}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({\sin k}^{2} \cdot \color{blue}{{t}^{3}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({t}^{3}\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \left(\ell \cdot \cos k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      5. cube-unmult N/A

         \[\leadsto \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{\log t \cdot 3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{\ell \cdot \ell} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      10. div-exp N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)} \]

      13. tan-quot N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot e^{\color{blue}{3 \cdot \log t - \log \ell \cdot 2}}} \]

   7. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {\left(\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}\right)}^{\color{blue}{-1}} \]

      2. div-inv N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{1}{\frac{\ell}{t \cdot t}}\right)}^{-1} \]

      3. clear-num N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{t \cdot t}{\ell}\right)}^{-1} \]

      4. associate-/l* N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}^{-1} \]

      5. associate-*r* N/A

         \[\leadsto {\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \frac{t}{\ell}\right)}^{-1} \]

      6. unpow-prod-down N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{-1}} \]

      7. inv-pow N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \frac{1}{\color{blue}{\frac{t}{\ell}}} \]

      8. clear-num N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \frac{\ell}{\color{blue}{t}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

   9. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{{\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \frac{\ell}{t}} \]

   if 1.5000000000000001e-12 < k < 1.84999999999999992e78

   1. Initial program 43.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.2%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 \cdot 1}{{k}^{2}} \cdot \frac{\color{blue}{{\ell}^{2}} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(2 \cdot \frac{1}{{k}^{2}}\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \frac{1}{{k}^{2}}\right), \color{blue}{\left(\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot 1}{{k}^{2}}\right), \left(\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{{k}^{2}}\right), \left(\frac{\color{blue}{{\ell}^{2}} \cdot \cos k}{t \cdot {\sin k}^{2}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left({k}^{2}\right)\right), \left(\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{t \cdot {\sin k}^{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{t \cdot {\sin k}^{2}}\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{\frac{\cos k}{{\sin k}^{2}}}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \color{blue}{\left(\frac{\cos k}{{\sin k}^{2}}\right)}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \left(\frac{\color{blue}{\cos k}}{{\sin k}^{2}}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \left(\frac{\cos \color{blue}{k}}{{\sin k}^{2}}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \left(\frac{\cos \color{blue}{k}}{{\sin k}^{2}}\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \mathsf{/.f64}\left(\cos k, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right) \]

      17. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right) \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right) \]

      19. sin-lowering-sin.f64 87.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right) \]

   7. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   if 1.84999999999999992e78 < k

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]
   7. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\right) \cdot \frac{2}{t}\right), t\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{k}{\frac{t}{k} \cdot t}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t}{k} \cdot t\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t \cdot t}{k}\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\sin k \cdot \tan k}\right), \left(\frac{2}{t}\right)\right)\right), t\right) \]

   8. Applied egg-rr 75.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{\frac{\ell}{\tan k}}{\sin k} \cdot \frac{2}{t}\right)}}{t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{e^{3 \cdot \log t + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\ell}{t} \cdot {\left(t \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right)}^{-1}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{+78}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{3 \cdot \log t\_m + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{l\_m}{t\_m} \cdot {\left(t\_m \cdot \frac{\sin k \cdot \tan k}{\frac{l\_m}{t\_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m}}{2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}} \cdot \left(\frac{2}{t\_m} \cdot \frac{\frac{l\_m}{\tan k}}{\sin k}\right)}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.8e-160)
    (* l_m (/ l_m (exp (+ (* 3.0 (log t_m)) (* 2.0 (log k))))))
    (if (<= k 6.2e-6)
      (* (/ l_m t_m) (pow (* t_m (/ (* (sin k) (tan k)) (/ l_m t_m))) -1.0))
      (/
       (*
        (/ (/ l_m t_m) (+ 2.0 (/ k (/ (* t_m t_m) k))))
        (* (/ 2.0 t_m) (/ (/ l_m (tan k)) (sin k))))
       t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.8e-160) {
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	} else if (k <= 6.2e-6) {
		tmp = (l_m / t_m) * pow((t_m * ((sin(k) * tan(k)) / (l_m / t_m))), -1.0);
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.8d-160) then
        tmp = l_m * (l_m / exp(((3.0d0 * log(t_m)) + (2.0d0 * log(k)))))
    else if (k <= 6.2d-6) then
        tmp = (l_m / t_m) * ((t_m * ((sin(k) * tan(k)) / (l_m / t_m))) ** (-1.0d0))
    else
        tmp = (((l_m / t_m) / (2.0d0 + (k / ((t_m * t_m) / k)))) * ((2.0d0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.8e-160) {
		tmp = l_m * (l_m / Math.exp(((3.0 * Math.log(t_m)) + (2.0 * Math.log(k)))));
	} else if (k <= 6.2e-6) {
		tmp = (l_m / t_m) * Math.pow((t_m * ((Math.sin(k) * Math.tan(k)) / (l_m / t_m))), -1.0);
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / Math.tan(k)) / Math.sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.8e-160:
		tmp = l_m * (l_m / math.exp(((3.0 * math.log(t_m)) + (2.0 * math.log(k)))))
	elif k <= 6.2e-6:
		tmp = (l_m / t_m) * math.pow((t_m * ((math.sin(k) * math.tan(k)) / (l_m / t_m))), -1.0)
	else:
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / math.tan(k)) / math.sin(k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.8e-160)
		tmp = Float64(l_m * Float64(l_m / exp(Float64(Float64(3.0 * log(t_m)) + Float64(2.0 * log(k))))));
	elseif (k <= 6.2e-6)
		tmp = Float64(Float64(l_m / t_m) * (Float64(t_m * Float64(Float64(sin(k) * tan(k)) / Float64(l_m / t_m))) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / t_m) / Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) * Float64(Float64(2.0 / t_m) * Float64(Float64(l_m / tan(k)) / sin(k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.8e-160)
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	elseif (k <= 6.2e-6)
		tmp = (l_m / t_m) * ((t_m * ((sin(k) * tan(k)) / (l_m / t_m))) ^ -1.0);
	else
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.8e-160], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.2e-6], N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Power[N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.7999999999999999e-160

   1. Initial program 64.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 63.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 64.4%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}\right)\right)\right) \]

       2. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot e^{\log t \cdot 3}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t}\right)\right)\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left({k}^{2} \cdot e^{\color{blue}{3 \cdot \log t}}\right)\right)\right) \]

       5. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(e^{\log k \cdot 2} \cdot e^{\color{blue}{3 \cdot \log t}}\right)\right)\right) \]

       6. prod-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(e^{\log k \cdot 2 + 3 \cdot \log t}\right)\right)\right) \]

       7. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\left(\log k \cdot 2 + 3 \cdot \log t\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log k \cdot 2\right), \left(3 \cdot \log t\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log k, 2\right), \left(3 \cdot \log t\right)\right)\right)\right)\right) \]

       10. log-lowering-log.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \left(3 \cdot \log t\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(3, \log t\right)\right)\right)\right)\right) \]

       12. log-lowering-log.f64 10.7%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 10.7%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{e^{\log k \cdot 2 + 3 \cdot \log t}}} \]

   if 1.7999999999999999e-160 < k < 6.1999999999999999e-6

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \cos k\right), \color{blue}{\left({t}^{3} \cdot {\sin k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \cos k\right), \left({\color{blue}{t}}^{3} \cdot {\sin k}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \cos k\right)\right), \left({t}^{\color{blue}{3}} \cdot {\sin k}^{2}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({t}^{3} \cdot {\sin k}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({\sin k}^{2} \cdot \color{blue}{{t}^{3}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({t}^{3}\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \left(\ell \cdot \cos k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      5. cube-unmult N/A

         \[\leadsto \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{\log t \cdot 3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{\ell \cdot \ell} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      10. div-exp N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)} \]

      13. tan-quot N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot e^{\color{blue}{3 \cdot \log t - \log \ell \cdot 2}}} \]

   7. Applied egg-rr 75.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {\left(\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}\right)}^{\color{blue}{-1}} \]

      2. div-inv N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{1}{\frac{\ell}{t \cdot t}}\right)}^{-1} \]

      3. clear-num N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{t \cdot t}{\ell}\right)}^{-1} \]

      4. associate-/l* N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}^{-1} \]

      5. associate-*r* N/A

         \[\leadsto {\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \frac{t}{\ell}\right)}^{-1} \]

      6. unpow-prod-down N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{\ell}\right)}^{-1}} \]

      7. inv-pow N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \frac{1}{\color{blue}{\frac{t}{\ell}}} \]

      8. clear-num N/A

         \[\leadsto {\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \frac{\ell}{\color{blue}{t}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

   9. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{{\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right)}^{-1} \cdot \frac{\ell}{t}} \]

   if 6.1999999999999999e-6 < k

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]
   7. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\right) \cdot \frac{2}{t}\right), t\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{k}{\frac{t}{k} \cdot t}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t}{k} \cdot t\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t \cdot t}{k}\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\sin k \cdot \tan k}\right), \left(\frac{2}{t}\right)\right)\right), t\right) \]

   8. Applied egg-rr 72.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{\frac{\ell}{\tan k}}{\sin k} \cdot \frac{2}{t}\right)}}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \frac{\ell}{e^{3 \cdot \log t + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\ell}{t} \cdot {\left(t \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{l\_m \cdot l\_m}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t\_m}}{t\_m}\right)}{l\_m} \cdot \frac{\tan k}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t\_m \cdot t\_m}\right)\right)} \cdot \frac{2}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.02e-47)
    (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l_m l_m)) (cos k)))
    (if (<= t_m 1e+94)
      (/
       2.0
       (*
        (/
         (* (* (sin k) (* t_m (* t_m t_m))) (+ 2.0 (/ (/ (* k k) t_m) t_m)))
         l_m)
        (/ (tan k) l_m)))
      (/
       (*
        (/
         (* (/ l_m t_m) (/ l_m k))
         (+ 2.0 (* k (* k (+ 0.3333333333333333 (/ 1.0 (* t_m t_m)))))))
        (/ 2.0 t_m))
       (* t_m k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.02e-47) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l_m * l_m)) / cos(k));
	} else if (t_m <= 1e+94) {
		tmp = 2.0 / ((((sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (tan(k) / l_m));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.02d-47) then
        tmp = 2.0d0 / (((k * (k * (t_m * (sin(k) ** 2.0d0)))) / (l_m * l_m)) / cos(k))
    else if (t_m <= 1d+94) then
        tmp = 2.0d0 / ((((sin(k) * (t_m * (t_m * t_m))) * (2.0d0 + (((k * k) / t_m) / t_m))) / l_m) * (tan(k) / l_m))
    else
        tmp = ((((l_m / t_m) * (l_m / k)) / (2.0d0 + (k * (k * (0.3333333333333333d0 + (1.0d0 / (t_m * t_m))))))) * (2.0d0 / t_m)) / (t_m * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.02e-47) {
		tmp = 2.0 / (((k * (k * (t_m * Math.pow(Math.sin(k), 2.0)))) / (l_m * l_m)) / Math.cos(k));
	} else if (t_m <= 1e+94) {
		tmp = 2.0 / ((((Math.sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (Math.tan(k) / l_m));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 1.02e-47:
		tmp = 2.0 / (((k * (k * (t_m * math.pow(math.sin(k), 2.0)))) / (l_m * l_m)) / math.cos(k))
	elif t_m <= 1e+94:
		tmp = 2.0 / ((((math.sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (math.tan(k) / l_m))
	else:
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.02e-47)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l_m * l_m)) / cos(k)));
	elseif (t_m <= 1e+94)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) * Float64(2.0 + Float64(Float64(Float64(k * k) / t_m) / t_m))) / l_m) * Float64(tan(k) / l_m)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(2.0 + Float64(k * Float64(k * Float64(0.3333333333333333 + Float64(1.0 / Float64(t_m * t_m))))))) * Float64(2.0 / t_m)) / Float64(t_m * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.02e-47)
		tmp = 2.0 / (((k * (k * (t_m * (sin(k) ^ 2.0)))) / (l_m * l_m)) / cos(k));
	elseif (t_m <= 1e+94)
		tmp = 2.0 / ((((sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (tan(k) / l_m));
	else
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.02e-47], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+94], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(k * N[(k * N[(0.3333333333333333 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.02000000000000002e-47

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sin k \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sin k \cdot \frac{1}{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot 1}{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot 1}{\frac{\ell \cdot \ell}{{t}^{3}}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot 1}{\frac{\ell \cdot \ell}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot 1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}} \cdot \frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}} \cdot \frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right), \left(\frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \left(\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}\right)\right), \left(\frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}\right)\right), \left(\frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \left({t}^{\left(\frac{3}{2}\right)}\right)\right)\right), \left(\frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      13. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right)\right)\right), \left(\frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \frac{3}{2}\right)\right)\right), \left(\frac{1}{\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \frac{3}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{\ell}{{t}^{\left(\frac{3}{2}\right)}}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \frac{3}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\ell, \left({t}^{\left(\frac{3}{2}\right)}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      17. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \frac{3}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \left(\frac{3}{2}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      18. metadata-eval 20.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \frac{3}{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\ell, \mathsf{pow.f64}\left(t, \frac{3}{2}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 20.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k}{\frac{\ell}{{t}^{1.5}}} \cdot \frac{1}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2}}\right), \color{blue}{\cos k}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos \color{blue}{k}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {\sin k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left({\sin k}^{2}\right)\right)\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, 2\right)\right)\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right), \left({\ell}^{2}\right)\right), \cos k\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right), \left(\ell \cdot \ell\right)\right), \cos k\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right)\right) \]

      13. cos-lowering-cos.f64 71.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right)\right) \]

   7. Simplified 71.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}} \]

   if 1.02000000000000002e-47 < t < 1e94

   1. Initial program 75.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t}}{t}\right)}{\ell} \cdot \frac{\tan k}{\ell}}} \]

   if 1e94 < t

   1. Initial program 63.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot t} + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot t} + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot t\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\left({k}^{2} \cdot t\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(t \cdot {k}^{2}\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2}\right)\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 60.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 60.1%

      \[\leadsto \frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\color{blue}{\ell}}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}}{\color{blue}{k}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}\right), \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 73.3%

      \[\leadsto \frac{2}{t \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}}{k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}}{k} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}}{k} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

       3. frac-times N/A

          \[\leadsto \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}\right) \cdot \frac{2}{t}}{\color{blue}{k \cdot t}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}\right) \cdot \frac{2}{t}\right), \color{blue}{\left(k \cdot t\right)}\right) \]

   11. Applied egg-rr 81.2%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{k \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t}}{t}\right)}{\ell} \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{3 \cdot \log t\_m + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{l\_m} \cdot \left(t\_m \cdot \frac{\sin k \cdot \tan k}{\frac{l\_m}{t\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m}}{2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}} \cdot \left(\frac{2}{t\_m} \cdot \frac{\frac{l\_m}{\tan k}}{\sin k}\right)}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.4e-161)
    (* l_m (/ l_m (exp (+ (* 3.0 (log t_m)) (* 2.0 (log k))))))
    (if (<= k 4.5e-6)
      (/ 1.0 (* (/ t_m l_m) (* t_m (/ (* (sin k) (tan k)) (/ l_m t_m)))))
      (/
       (*
        (/ (/ l_m t_m) (+ 2.0 (/ k (/ (* t_m t_m) k))))
        (* (/ 2.0 t_m) (/ (/ l_m (tan k)) (sin k))))
       t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.4e-161) {
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	} else if (k <= 4.5e-6) {
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((sin(k) * tan(k)) / (l_m / t_m))));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.4d-161) then
        tmp = l_m * (l_m / exp(((3.0d0 * log(t_m)) + (2.0d0 * log(k)))))
    else if (k <= 4.5d-6) then
        tmp = 1.0d0 / ((t_m / l_m) * (t_m * ((sin(k) * tan(k)) / (l_m / t_m))))
    else
        tmp = (((l_m / t_m) / (2.0d0 + (k / ((t_m * t_m) / k)))) * ((2.0d0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.4e-161) {
		tmp = l_m * (l_m / Math.exp(((3.0 * Math.log(t_m)) + (2.0 * Math.log(k)))));
	} else if (k <= 4.5e-6) {
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((Math.sin(k) * Math.tan(k)) / (l_m / t_m))));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / Math.tan(k)) / Math.sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 3.4e-161:
		tmp = l_m * (l_m / math.exp(((3.0 * math.log(t_m)) + (2.0 * math.log(k)))))
	elif k <= 4.5e-6:
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((math.sin(k) * math.tan(k)) / (l_m / t_m))))
	else:
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / math.tan(k)) / math.sin(k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.4e-161)
		tmp = Float64(l_m * Float64(l_m / exp(Float64(Float64(3.0 * log(t_m)) + Float64(2.0 * log(k))))));
	elseif (k <= 4.5e-6)
		tmp = Float64(1.0 / Float64(Float64(t_m / l_m) * Float64(t_m * Float64(Float64(sin(k) * tan(k)) / Float64(l_m / t_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / t_m) / Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) * Float64(Float64(2.0 / t_m) * Float64(Float64(l_m / tan(k)) / sin(k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 3.4e-161)
		tmp = l_m * (l_m / exp(((3.0 * log(t_m)) + (2.0 * log(k)))));
	elseif (k <= 4.5e-6)
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((sin(k) * tan(k)) / (l_m / t_m))));
	else
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.4e-161], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.5e-6], N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.39999999999999982e-161

   1. Initial program 64.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 63.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 64.4%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}\right)\right)\right) \]

       2. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot e^{\log t \cdot 3}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t}\right)\right)\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left({k}^{2} \cdot e^{\color{blue}{3 \cdot \log t}}\right)\right)\right) \]

       5. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(e^{\log k \cdot 2} \cdot e^{\color{blue}{3 \cdot \log t}}\right)\right)\right) \]

       6. prod-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(e^{\log k \cdot 2 + 3 \cdot \log t}\right)\right)\right) \]

       7. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\left(\log k \cdot 2 + 3 \cdot \log t\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log k \cdot 2\right), \left(3 \cdot \log t\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log k, 2\right), \left(3 \cdot \log t\right)\right)\right)\right)\right) \]

       10. log-lowering-log.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \left(3 \cdot \log t\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(3, \log t\right)\right)\right)\right)\right) \]

       12. log-lowering-log.f64 10.7%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(3, \mathsf{log.f64}\left(t\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 10.7%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{e^{\log k \cdot 2 + 3 \cdot \log t}}} \]

   if 3.39999999999999982e-161 < k < 4.50000000000000011e-6

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \cos k\right), \color{blue}{\left({t}^{3} \cdot {\sin k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \cos k\right), \left({\color{blue}{t}}^{3} \cdot {\sin k}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \cos k\right)\right), \left({t}^{\color{blue}{3}} \cdot {\sin k}^{2}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({t}^{3} \cdot {\sin k}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({\sin k}^{2} \cdot \color{blue}{{t}^{3}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({t}^{3}\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \left(\ell \cdot \cos k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      5. cube-unmult N/A

         \[\leadsto \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{\log t \cdot 3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{\ell \cdot \ell} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      10. div-exp N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)} \]

      13. tan-quot N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot e^{\color{blue}{3 \cdot \log t - \log \ell \cdot 2}}} \]

   7. Applied egg-rr 75.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{1}{\frac{\ell}{t \cdot t}}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{t \cdot t}{\color{blue}{\ell}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right), t\right), \left(\frac{\color{blue}{t}}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sin k \cdot \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 84.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right), \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 84.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \frac{t}{\ell}}} \]

   if 4.50000000000000011e-6 < k

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]
   7. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\right) \cdot \frac{2}{t}\right), t\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{k}{\frac{t}{k} \cdot t}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t}{k} \cdot t\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t \cdot t}{k}\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\sin k \cdot \tan k}\right), \left(\frac{2}{t}\right)\right)\right), t\right) \]

   8. Applied egg-rr 72.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{\frac{\ell}{\tan k}}{\sin k} \cdot \frac{2}{t}\right)}}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \frac{\ell}{e^{3 \cdot \log t + 2 \cdot \log k}}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{l\_m} \cdot \left(t\_m \cdot \frac{\sin k \cdot \tan k}{\frac{l\_m}{t\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m}}{2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}} \cdot \left(\frac{2}{t\_m} \cdot \frac{\frac{l\_m}{\tan k}}{\sin k}\right)}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.05e-158)
    (/ (* (/ l_m t_m) (/ l_m k)) (* k (* t_m t_m)))
    (if (<= k 4.2e-6)
      (/ 1.0 (* (/ t_m l_m) (* t_m (/ (* (sin k) (tan k)) (/ l_m t_m)))))
      (/
       (*
        (/ (/ l_m t_m) (+ 2.0 (/ k (/ (* t_m t_m) k))))
        (* (/ 2.0 t_m) (/ (/ l_m (tan k)) (sin k))))
       t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.05e-158) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else if (k <= 4.2e-6) {
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((sin(k) * tan(k)) / (l_m / t_m))));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.05d-158) then
        tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
    else if (k <= 4.2d-6) then
        tmp = 1.0d0 / ((t_m / l_m) * (t_m * ((sin(k) * tan(k)) / (l_m / t_m))))
    else
        tmp = (((l_m / t_m) / (2.0d0 + (k / ((t_m * t_m) / k)))) * ((2.0d0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.05e-158) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else if (k <= 4.2e-6) {
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((Math.sin(k) * Math.tan(k)) / (l_m / t_m))));
	} else {
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / Math.tan(k)) / Math.sin(k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.05e-158:
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
	elif k <= 4.2e-6:
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((math.sin(k) * math.tan(k)) / (l_m / t_m))))
	else:
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / math.tan(k)) / math.sin(k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.05e-158)
		tmp = Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(k * Float64(t_m * t_m)));
	elseif (k <= 4.2e-6)
		tmp = Float64(1.0 / Float64(Float64(t_m / l_m) * Float64(t_m * Float64(Float64(sin(k) * tan(k)) / Float64(l_m / t_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / t_m) / Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) * Float64(Float64(2.0 / t_m) * Float64(Float64(l_m / tan(k)) / sin(k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.05e-158)
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	elseif (k <= 4.2e-6)
		tmp = 1.0 / ((t_m / l_m) * (t_m * ((sin(k) * tan(k)) / (l_m / t_m))));
	else
		tmp = (((l_m / t_m) / (2.0 + (k / ((t_m * t_m) / k)))) * ((2.0 / t_m) * ((l_m / tan(k)) / sin(k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.05e-158], N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.2e-6], N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.04999999999999996e-158

   1. Initial program 63.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{\ell}{k}\right)\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{k}\right)\right), \left(\left(\color{blue}{t} \cdot t\right) \cdot k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \left(\left(t \cdot \color{blue}{t}\right) \cdot k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{k}\right)\right) \]

      10. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right) \]

   9. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\left(t \cdot t\right) \cdot k}} \]

   if 1.04999999999999996e-158 < k < 4.1999999999999996e-6

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \cos k\right), \color{blue}{\left({t}^{3} \cdot {\sin k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \cos k\right), \left({\color{blue}{t}}^{3} \cdot {\sin k}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \cos k\right)\right), \left({t}^{\color{blue}{3}} \cdot {\sin k}^{2}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({t}^{3} \cdot {\sin k}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({\sin k}^{2} \cdot \color{blue}{{t}^{3}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({t}^{3}\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \left(\ell \cdot \cos k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      5. cube-unmult N/A

         \[\leadsto \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{\log t \cdot 3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{\ell \cdot \ell} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      10. div-exp N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)} \]

      13. tan-quot N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot e^{\color{blue}{3 \cdot \log t - \log \ell \cdot 2}}} \]

   7. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{1}{\frac{\ell}{t \cdot t}}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{t \cdot t}{\color{blue}{\ell}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right), t\right), \left(\frac{\color{blue}{t}}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sin k \cdot \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 84.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right), \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 84.5%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \frac{t}{\ell}}} \]

   if 4.1999999999999996e-6 < k

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]
   7. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\right) \cdot \frac{2}{t}\right), t\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}} \cdot \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \left(\frac{k}{\frac{t}{k} \cdot t}\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t}{k} \cdot t\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{t \cdot t}{k}\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \left(\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{t}\right)\right), t\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\sin k \cdot \tan k}\right), \left(\frac{2}{t}\right)\right)\right), t\right) \]

   8. Applied egg-rr 72.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{\frac{\ell}{\tan k}}{\sin k} \cdot \frac{2}{t}\right)}}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{\frac{t \cdot t}{k}}} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{\tan k}}{\sin k}\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{l\_m} \cdot \left(t\_m \cdot \frac{t\_2}{\frac{l\_m}{t\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m} \cdot \frac{l\_m \cdot \frac{l\_m}{t\_m}}{\left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right) \cdot t\_2}}{t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (sin k) (tan k))))
   (*
    t_s
    (if (<= k 1.02e-158)
      (/ (* (/ l_m t_m) (/ l_m k)) (* k (* t_m t_m)))
      (if (<= k 5.5e-6)
        (/ 1.0 (* (/ t_m l_m) (* t_m (/ t_2 (/ l_m t_m)))))
        (/
         (*
          (/ 2.0 t_m)
          (/ (* l_m (/ l_m t_m)) (* (+ 2.0 (/ (/ k t_m) (/ t_m k))) t_2)))
         t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.02e-158) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else if (k <= 5.5e-6) {
		tmp = 1.0 / ((t_m / l_m) * (t_m * (t_2 / (l_m / t_m))));
	} else {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / ((2.0 + ((k / t_m) / (t_m / k))) * t_2))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) * tan(k)
    if (k <= 1.02d-158) then
        tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
    else if (k <= 5.5d-6) then
        tmp = 1.0d0 / ((t_m / l_m) * (t_m * (t_2 / (l_m / t_m))))
    else
        tmp = ((2.0d0 / t_m) * ((l_m * (l_m / t_m)) / ((2.0d0 + ((k / t_m) / (t_m / k))) * t_2))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.02e-158) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else if (k <= 5.5e-6) {
		tmp = 1.0 / ((t_m / l_m) * (t_m * (t_2 / (l_m / t_m))));
	} else {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / ((2.0 + ((k / t_m) / (t_m / k))) * t_2))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 1.02e-158:
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
	elif k <= 5.5e-6:
		tmp = 1.0 / ((t_m / l_m) * (t_m * (t_2 / (l_m / t_m))))
	else:
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / ((2.0 + ((k / t_m) / (t_m / k))) * t_2))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.02e-158)
		tmp = Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(k * Float64(t_m * t_m)));
	elseif (k <= 5.5e-6)
		tmp = Float64(1.0 / Float64(Float64(t_m / l_m) * Float64(t_m * Float64(t_2 / Float64(l_m / t_m)))));
	else
		tmp = Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(l_m * Float64(l_m / t_m)) / Float64(Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))) * t_2))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 1.02e-158)
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	elseif (k <= 5.5e-6)
		tmp = 1.0 / ((t_m / l_m) * (t_m * (t_2 / (l_m / t_m))));
	else
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / ((2.0 + ((k / t_m) / (t_m / k))) * t_2))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.02e-158], N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.5e-6], N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m * N[(t$95$2 / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0199999999999999e-158

   1. Initial program 63.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{\ell}{k}\right)\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{k}\right)\right), \left(\left(\color{blue}{t} \cdot t\right) \cdot k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \left(\left(t \cdot \color{blue}{t}\right) \cdot k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{k}\right)\right) \]

      10. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right) \]

   9. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\left(t \cdot t\right) \cdot k}} \]

   if 1.0199999999999999e-158 < k < 5.4999999999999999e-6

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \cos k\right), \color{blue}{\left({t}^{3} \cdot {\sin k}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \cos k\right), \left({\color{blue}{t}}^{3} \cdot {\sin k}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \cos k\right)\right), \left(\color{blue}{{t}^{3}} \cdot {\sin k}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \cos k\right)\right), \left({t}^{\color{blue}{3}} \cdot {\sin k}^{2}\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({t}^{3} \cdot {\sin k}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({\sin k}^{2} \cdot \color{blue}{{t}^{3}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\left({t}^{3}\right)}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left({\color{blue}{t}}^{3}\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({t}^{3}\right)\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \left(\ell \cdot \cos k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      5. cube-unmult N/A

         \[\leadsto \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{\log t \cdot 3}}{\ell \cdot \ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{\ell \cdot \ell} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{{\ell}^{2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      9. pow-to-exp N/A

         \[\leadsto \frac{1}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\cos k}} \]

      10. div-exp N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)} \]

      13. tan-quot N/A

          \[\leadsto \frac{1}{e^{3 \cdot \log t - \log \ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}}} \]

      15. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot e^{\color{blue}{3 \cdot \log t - \log \ell \cdot 2}}} \]

   7. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t \cdot t}}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{1}{\frac{\ell}{t \cdot t}}}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{t \cdot t}{\color{blue}{\ell}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right), t\right), \left(\frac{\color{blue}{t}}{\ell}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sin k \cdot \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin k, \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      10. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\ell}{t}\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right), \left(\frac{t}{\ell}\right)\right)\right) \]

      12. /-lowering-/.f64 84.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), t\right), \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 84.5%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot t\right) \cdot \frac{t}{\ell}}} \]

   if 5.4999999999999999e-6 < k

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{\ell \cdot \frac{\ell}{t}}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{2}{t\_m} \cdot \frac{l\_m \cdot \frac{l\_m}{t\_m}}{2 \cdot \left(k \cdot k\right)}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t\_m}}{t\_m}\right)}{l\_m} \cdot \frac{\tan k}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t\_m \cdot t\_m}\right)\right)} \cdot \frac{2}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-114)
    (/ (* (/ 2.0 t_m) (/ (* l_m (/ l_m t_m)) (* 2.0 (* k k)))) t_m)
    (if (<= t_m 2.55e+94)
      (/
       2.0
       (*
        (/
         (* (* (sin k) (* t_m (* t_m t_m))) (+ 2.0 (/ (/ (* k k) t_m) t_m)))
         l_m)
        (/ (tan k) l_m)))
      (/
       (*
        (/
         (* (/ l_m t_m) (/ l_m k))
         (+ 2.0 (* k (* k (+ 0.3333333333333333 (/ 1.0 (* t_m t_m)))))))
        (/ 2.0 t_m))
       (* t_m k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 3.4e-114) {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	} else if (t_m <= 2.55e+94) {
		tmp = 2.0 / ((((sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (tan(k) / l_m));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.4d-114) then
        tmp = ((2.0d0 / t_m) * ((l_m * (l_m / t_m)) / (2.0d0 * (k * k)))) / t_m
    else if (t_m <= 2.55d+94) then
        tmp = 2.0d0 / ((((sin(k) * (t_m * (t_m * t_m))) * (2.0d0 + (((k * k) / t_m) / t_m))) / l_m) * (tan(k) / l_m))
    else
        tmp = ((((l_m / t_m) * (l_m / k)) / (2.0d0 + (k * (k * (0.3333333333333333d0 + (1.0d0 / (t_m * t_m))))))) * (2.0d0 / t_m)) / (t_m * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 3.4e-114) {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	} else if (t_m <= 2.55e+94) {
		tmp = 2.0 / ((((Math.sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (Math.tan(k) / l_m));
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 3.4e-114:
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m
	elif t_m <= 2.55e+94:
		tmp = 2.0 / ((((math.sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (math.tan(k) / l_m))
	else:
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 3.4e-114)
		tmp = Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(l_m * Float64(l_m / t_m)) / Float64(2.0 * Float64(k * k)))) / t_m);
	elseif (t_m <= 2.55e+94)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) * Float64(2.0 + Float64(Float64(Float64(k * k) / t_m) / t_m))) / l_m) * Float64(tan(k) / l_m)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(2.0 + Float64(k * Float64(k * Float64(0.3333333333333333 + Float64(1.0 / Float64(t_m * t_m))))))) * Float64(2.0 / t_m)) / Float64(t_m * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 3.4e-114)
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	elseif (t_m <= 2.55e+94)
		tmp = 2.0 / ((((sin(k) * (t_m * (t_m * t_m))) * (2.0 + (((k * k) / t_m) / t_m))) / l_m) * (tan(k) / l_m));
	else
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-114], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+94], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(k * N[(k * N[(0.3333333333333333 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.39999999999999981e-114

   1. Initial program 53.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \color{blue}{\left(2 \cdot {k}^{2}\right)}\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \left({k}^{2}\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      3. *-lowering-*.f64 70.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

   9. Simplified 70.8%

      \[\leadsto \frac{\frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{2 \cdot \left(k \cdot k\right)}} \cdot \frac{2}{t}}{t} \]

   if 3.39999999999999981e-114 < t < 2.5500000000000002e94

   1. Initial program 77.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   4. Applied egg-rr 93.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t}}{t}\right)}{\ell} \cdot \frac{\tan k}{\ell}}} \]

   if 2.5500000000000002e94 < t

   1. Initial program 63.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot t} + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot t} + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot t\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\left({k}^{2} \cdot t\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(t \cdot {k}^{2}\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2}\right)\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 60.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 60.1%

      \[\leadsto \frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\color{blue}{\ell}}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}}{\color{blue}{k}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}\right), \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 73.3%

      \[\leadsto \frac{2}{t \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}}{k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}}{k} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}}{k} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

       3. frac-times N/A

          \[\leadsto \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}\right) \cdot \frac{2}{t}}{\color{blue}{k \cdot t}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}\right) \cdot \frac{2}{t}\right), \color{blue}{\left(k \cdot t\right)}\right) \]

   11. Applied egg-rr 81.2%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{k \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{\ell \cdot \frac{\ell}{t}}{2 \cdot \left(k \cdot k\right)}}{t}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t}}{t}\right)}{\ell} \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{2}{t\_m} \cdot \frac{l\_m \cdot \frac{l\_m}{t\_m}}{2 \cdot \left(k \cdot k\right)}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t\_m \cdot t\_m}\right)\right)} \cdot \frac{2}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.22e-118)
    (/ (* (/ 2.0 t_m) (/ (* l_m (/ l_m t_m)) (* 2.0 (* k k)))) t_m)
    (/
     (*
      (/
       (* (/ l_m t_m) (/ l_m k))
       (+ 2.0 (* k (* k (+ 0.3333333333333333 (/ 1.0 (* t_m t_m)))))))
      (/ 2.0 t_m))
     (* t_m k)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.22e-118) {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.22d-118) then
        tmp = ((2.0d0 / t_m) * ((l_m * (l_m / t_m)) / (2.0d0 * (k * k)))) / t_m
    else
        tmp = ((((l_m / t_m) * (l_m / k)) / (2.0d0 + (k * (k * (0.3333333333333333d0 + (1.0d0 / (t_m * t_m))))))) * (2.0d0 / t_m)) / (t_m * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.22e-118) {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	} else {
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 1.22e-118:
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m
	else:
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.22e-118)
		tmp = Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(l_m * Float64(l_m / t_m)) / Float64(2.0 * Float64(k * k)))) / t_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(2.0 + Float64(k * Float64(k * Float64(0.3333333333333333 + Float64(1.0 / Float64(t_m * t_m))))))) * Float64(2.0 / t_m)) / Float64(t_m * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.22e-118)
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	else
		tmp = ((((l_m / t_m) * (l_m / k)) / (2.0 + (k * (k * (0.3333333333333333 + (1.0 / (t_m * t_m))))))) * (2.0 / t_m)) / (t_m * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.22e-118], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(k * N[(k * N[(0.3333333333333333 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2200000000000001e-118

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \color{blue}{\left(2 \cdot {k}^{2}\right)}\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \left({k}^{2}\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      3. *-lowering-*.f64 70.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

   9. Simplified 70.7%

      \[\leadsto \frac{\frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{2 \cdot \left(k \cdot k\right)}} \cdot \frac{2}{t}}{t} \]

   if 1.2200000000000001e-118 < t

   1. Initial program 70.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \color{blue}{\left({k}^{2} \cdot \left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot t} + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot t} + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot t\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\left({k}^{2} \cdot t\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(t \cdot {k}^{2}\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2}\right)\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{{t}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 66.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 66.3%

      \[\leadsto \frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\color{blue}{\ell}}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}}{\color{blue}{k}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{2 \cdot t + \left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)}\right), \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 76.1%

      \[\leadsto \frac{2}{t \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)}}{k}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}}{k} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}}{k} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

       3. frac-times N/A

          \[\leadsto \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}\right) \cdot \frac{2}{t}}{\color{blue}{k \cdot t}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(2 + \left(k \cdot k\right) \cdot \left(\frac{1}{3} + \frac{1}{t \cdot t}\right)\right)}\right) \cdot \frac{2}{t}\right), \color{blue}{\left(k \cdot t\right)}\right) \]

   11. Applied egg-rr 80.9%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{\ell \cdot \frac{\ell}{t}}{2 \cdot \left(k \cdot k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{2 + k \cdot \left(k \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right)\right)} \cdot \frac{2}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m} \cdot \frac{l\_m \cdot \frac{l\_m}{t\_m}}{2 \cdot \left(k \cdot k\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.2e-36)
    (/ (* (/ l_m t_m) (/ l_m k)) (* k (* t_m t_m)))
    (/ (* (/ 2.0 t_m) (/ (* l_m (/ l_m t_m)) (* 2.0 (* k k)))) t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.2e-36) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d-36) then
        tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
    else
        tmp = ((2.0d0 / t_m) * ((l_m * (l_m / t_m)) / (2.0d0 * (k * k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.2e-36) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else {
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.2e-36:
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
	else:
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.2e-36)
		tmp = Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(k * Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(l_m * Float64(l_m / t_m)) / Float64(2.0 * Float64(k * k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.2e-36)
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	else
		tmp = ((2.0 / t_m) * ((l_m * (l_m / t_m)) / (2.0 * (k * k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.2e-36], N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.2e-36

   1. Initial program 63.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 64.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{\ell}{k}\right)\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{k}\right)\right), \left(\left(\color{blue}{t} \cdot t\right) \cdot k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \left(\left(t \cdot \color{blue}{t}\right) \cdot k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{k}\right)\right) \]

      10. *-lowering-*.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right) \]

   9. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\left(t \cdot t\right) \cdot k}} \]

   if 1.2e-36 < k

   1. Initial program 46.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \color{blue}{\left(2 \cdot {k}^{2}\right)}\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \left({k}^{2}\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      3. *-lowering-*.f64 68.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

   9. Simplified 68.3%

      \[\leadsto \frac{\frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{2 \cdot \left(k \cdot k\right)}} \cdot \frac{2}{t}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{\ell \cdot \frac{\ell}{t}}{2 \cdot \left(k \cdot k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.4% accurate, 19.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 720000000:\\ \;\;\;\;\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m} \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot 0.5}{t\_m \cdot \left(k \cdot k\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 720000000.0)
    (/ (* (/ l_m t_m) (/ l_m k)) (* k (* t_m t_m)))
    (/ (* (/ 2.0 t_m) (/ (* (* l_m l_m) 0.5) (* t_m (* k k)))) t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 720000000.0) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else {
		tmp = ((2.0 / t_m) * (((l_m * l_m) * 0.5) / (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 720000000.0d0) then
        tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
    else
        tmp = ((2.0d0 / t_m) * (((l_m * l_m) * 0.5d0) / (t_m * (k * k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 720000000.0) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else {
		tmp = ((2.0 / t_m) * (((l_m * l_m) * 0.5) / (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 720000000.0:
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
	else:
		tmp = ((2.0 / t_m) * (((l_m * l_m) * 0.5) / (t_m * (k * k)))) / t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 720000000.0)
		tmp = Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(k * Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(Float64(l_m * l_m) * 0.5) / Float64(t_m * Float64(k * k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 720000000.0)
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	else
		tmp = ((2.0 / t_m) * (((l_m * l_m) * 0.5) / (t_m * (k * k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 720000000.0], N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.2e8

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{\ell}{k}\right)\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{k}\right)\right), \left(\left(\color{blue}{t} \cdot t\right) \cdot k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \left(\left(t \cdot \color{blue}{t}\right) \cdot k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{k}\right)\right) \]

      10. *-lowering-*.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right) \]

   9. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\left(t \cdot t\right) \cdot k}} \]

   if 7.2e8 < k

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\frac{2}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \color{blue}{\frac{2}{t \cdot t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{\frac{2}{t}}{\color{blue}{t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}}{\color{blue}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \frac{2}{t}\right), \color{blue}{t}\right) \]

   6. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{2}{t}}{t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(2, t\right)\right), t\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{2} \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {\ell}^{2}\right), \left({k}^{2} \cdot t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\ell}^{2}\right)\right), \left({k}^{2} \cdot t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\ell \cdot \ell\right)\right), \left({k}^{2} \cdot t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{2} \cdot t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

      8. *-lowering-*.f64 64.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \mathsf{/.f64}\left(2, t\right)\right), t\right) \]

   9. Simplified 64.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t}} \cdot \frac{2}{t}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 720000000:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{\left(\ell \cdot \ell\right) \cdot 0.5}{t \cdot \left(k \cdot k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.4% accurate, 23.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{k}}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.1e+35)
    (/ (* (/ l_m t_m) (/ l_m k)) (* k (* t_m t_m)))
    (* l_m (/ l_m (* t_m (* t_m (* t_m (* k k)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.1e+35) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else {
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d+35) then
        tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
    else
        tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.1e+35) {
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	} else {
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 3.1e+35:
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m))
	else:
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.1e+35)
		tmp = Float64(Float64(Float64(l_m / t_m) * Float64(l_m / k)) / Float64(k * Float64(t_m * t_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 3.1e+35)
		tmp = ((l_m / t_m) * (l_m / k)) / (k * (t_m * t_m));
	else
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.1e+35], N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999987e35

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 63.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{\ell}{k}\right)\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{k}\right)\right), \left(\left(\color{blue}{t} \cdot t\right) \cdot k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \left(\left(t \cdot \color{blue}{t}\right) \cdot k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{k}\right)\right) \]

      10. *-lowering-*.f64 74.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right) \]

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\left(t \cdot t\right) \cdot k}} \]

   if 3.09999999999999987e35 < k

   1. Initial program 52.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 55.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 57.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 55.7%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), t\right), t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

       7. *-lowering-*.f64 63.6%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), t\right), t\right)\right)\right) \]

   11. Applied egg-rr 63.6%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-159}:\\ \;\;\;\;l\_m \cdot \frac{\frac{l\_m}{k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 4e-159)
    (* l_m (/ (/ l_m (* k (* t_m (* t_m t_m)))) k))
    (* l_m (/ l_m (* t_m (* t_m (* t_m (* k k)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 4e-159) {
		tmp = l_m * ((l_m / (k * (t_m * (t_m * t_m)))) / k);
	} else {
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4d-159) then
        tmp = l_m * ((l_m / (k * (t_m * (t_m * t_m)))) / k)
    else
        tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 4e-159) {
		tmp = l_m * ((l_m / (k * (t_m * (t_m * t_m)))) / k);
	} else {
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 4e-159:
		tmp = l_m * ((l_m / (k * (t_m * (t_m * t_m)))) / k)
	else:
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 4e-159)
		tmp = Float64(l_m * Float64(Float64(l_m / Float64(k * Float64(t_m * Float64(t_m * t_m)))) / k));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 4e-159)
		tmp = l_m * ((l_m / (k * (t_m * (t_m * t_m)))) / k);
	else
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 4e-159], N[(l$95$m * N[(N[(l$95$m / N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.99999999999999995e-159

   1. Initial program 63.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 64.3%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \color{blue}{\ell} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\ell}\right) \]

       3. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}\right), \ell\right) \]

       4. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot e^{\log t \cdot 3}}\right), \ell\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot k\right) \cdot e^{3 \cdot \log t}}\right), \ell\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(k \cdot e^{3 \cdot \log t}\right)}\right), \ell\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot e^{3 \cdot \log t}\right) \cdot k}\right), \ell\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot e^{3 \cdot \log t}}}{k}\right), \ell\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot e^{3 \cdot \log t}}\right), k\right), \ell\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot e^{3 \cdot \log t}\right)\right), k\right), \ell\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot e^{\log t \cdot 3}\right)\right), k\right), \ell\right) \]

       12. pow-to-exp N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot {t}^{3}\right)\right), k\right), \ell\right) \]

       13. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), k\right), \ell\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), k\right), \ell\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), k\right), \ell\right) \]

       16. *-lowering-*.f64 73.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), k\right), \ell\right) \]

   11. Applied egg-rr 73.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k} \cdot \ell} \]

   if 3.99999999999999995e-159 < k

   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. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 56.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), t\right), t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

       7. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), t\right), t\right)\right)\right) \]

   11. Applied egg-rr 61.2%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{l\_m}{k} \cdot \frac{l\_m}{k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.8e-159)
    (* (/ l_m k) (/ l_m (* k (* t_m (* t_m t_m)))))
    (* l_m (/ l_m (* t_m (* t_m (* t_m (* k k)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 6.8e-159) {
		tmp = (l_m / k) * (l_m / (k * (t_m * (t_m * t_m))));
	} else {
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.8d-159) then
        tmp = (l_m / k) * (l_m / (k * (t_m * (t_m * t_m))))
    else
        tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 6.8e-159) {
		tmp = (l_m / k) * (l_m / (k * (t_m * (t_m * t_m))));
	} else {
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 6.8e-159:
		tmp = (l_m / k) * (l_m / (k * (t_m * (t_m * t_m))))
	else:
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 6.8e-159)
		tmp = Float64(Float64(l_m / k) * Float64(l_m / Float64(k * Float64(t_m * Float64(t_m * t_m)))));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 6.8e-159)
		tmp = (l_m / k) * (l_m / (k * (t_m * (t_m * t_m))));
	else
		tmp = l_m * (l_m / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 6.8e-159], N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.79999999999999967e-159

   1. Initial program 63.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \color{blue}{k}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 72.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   7. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{k}} \]

   if 6.79999999999999967e-159 < k

   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. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 55.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 56.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   8. Step-by-step derivation

      1. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), t\right), t\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

       7. *-lowering-*.f64 61.2%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), t\right), t\right)\right)\right) \]

   11. Applied egg-rr 61.2%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 61.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* l_m (/ l_m (* t_m (* t_m (* t_m (* k k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / (t_m * (t_m * (t_m * (k * k))))));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (l_m * (l_m / (t_m * (t_m * (t_m * (k * k))))))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / (t_m * (t_m * (t_m * (k * k))))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / (t_m * (t_m * (t_m * (k * k))))))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k)))))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / (t_m * (t_m * (t_m * (k * k))))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 61.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

5. Simplified 61.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 60.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

7. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
8. Step-by-step derivation

   1. frac-times N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   2. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 61.2%

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 61.2%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), t\right), t\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), t\right), t\right)\right)\right) \]

    7. *-lowering-*.f64 67.3%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), t\right), t\right)\right)\right) \]

11. Applied egg-rr 67.3%

    \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right) \cdot t}} \]

12. Final simplification 67.3%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \]
13. Add Preprocessing

Alternative 21: 55.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{\left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot k\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* l_m (/ l_m (* (* t_m (* t_m t_m)) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((t_m * (t_m * t_m)) * (k * k))));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (l_m * (l_m / ((t_m * (t_m * t_m)) * (k * k))))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((t_m * (t_m * t_m)) * (k * k))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / ((t_m * (t_m * t_m)) * (k * k))))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(Float64(t_m * Float64(t_m * t_m)) * Float64(k * k)))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / ((t_m * (t_m * t_m)) * (k * k))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 61.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

5. Simplified 61.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 60.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

7. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
8. Step-by-step derivation

   1. frac-times N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   2. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 61.2%

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 61.2%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

10. Final simplification 61.2%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))