Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 88.3%

Time: 17.3s

Alternatives: 12

Speedup: 23.4×

• 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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 1 + \left(k\_m \cdot k\_m\right) \cdot \left(0.16666666666666666 + \left(k\_m \cdot k\_m\right) \cdot 0.08611111111111111\right)\\ \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 2.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k\_m \leq 0.95:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot t\right)\right) \cdot t\_1\right) + t\_1 \cdot {k\_m}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1
         (+
          1.0
          (*
           (* k_m k_m)
           (+ 0.16666666666666666 (* (* k_m k_m) 0.08611111111111111))))))
   (if (<= k_m 1.25e-139)
     (* l (/ l (* (* t k_m) (* t (* t k_m)))))
     (if (<= k_m 2.2e-79)
       (* (/ l (* t (* t (* k_m k_m)))) (/ l t))
       (if (<= k_m 0.95)
         (/
          2.0
          (*
           (/ t l)
           (/
            (+ (* 2.0 (* (* (* k_m k_m) (* t t)) t_1)) (* t_1 (pow k_m 4.0)))
            l)))
         (/
          2.0
          (*
           (/ (* k_m (+ 0.5 (* (cos (* k_m 2.0)) -0.5))) l)
           (/ (* t (/ k_m (cos k_m))) l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 1.25e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 2.2e-79) {
		tmp = (l / (t * (t * (k_m * k_m)))) * (l / t);
	} else if (k_m <= 0.95) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * pow(k_m, 4.0))) / l));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / cos(k_m))) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + ((k_m * k_m) * (0.16666666666666666d0 + ((k_m * k_m) * 0.08611111111111111d0)))
    if (k_m <= 1.25d-139) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else if (k_m <= 2.2d-79) then
        tmp = (l / (t * (t * (k_m * k_m)))) * (l / t)
    else if (k_m <= 0.95d0) then
        tmp = 2.0d0 / ((t / l) * (((2.0d0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * (k_m ** 4.0d0))) / l))
    else
        tmp = 2.0d0 / (((k_m * (0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0)))) / l) * ((t * (k_m / cos(k_m))) / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 1.25e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 2.2e-79) {
		tmp = (l / (t * (t * (k_m * k_m)))) * (l / t);
	} else if (k_m <= 0.95) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * Math.pow(k_m, 4.0))) / l));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (Math.cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / Math.cos(k_m))) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)))
	tmp = 0
	if k_m <= 1.25e-139:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	elif k_m <= 2.2e-79:
		tmp = (l / (t * (t * (k_m * k_m)))) * (l / t)
	elif k_m <= 0.95:
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * math.pow(k_m, 4.0))) / l))
	else:
		tmp = 2.0 / (((k_m * (0.5 + (math.cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / math.cos(k_m))) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(1.0 + Float64(Float64(k_m * k_m) * Float64(0.16666666666666666 + Float64(Float64(k_m * k_m) * 0.08611111111111111))))
	tmp = 0.0
	if (k_m <= 1.25e-139)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	elseif (k_m <= 2.2e-79)
		tmp = Float64(Float64(l / Float64(t * Float64(t * Float64(k_m * k_m)))) * Float64(l / t));
	elseif (k_m <= 0.95)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(2.0 * Float64(Float64(Float64(k_m * k_m) * Float64(t * t)) * t_1)) + Float64(t_1 * (k_m ^ 4.0))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))) / l) * Float64(Float64(t * Float64(k_m / cos(k_m))) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	tmp = 0.0;
	if (k_m <= 1.25e-139)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	elseif (k_m <= 2.2e-79)
		tmp = (l / (t * (t * (k_m * k_m)))) * (l / t);
	elseif (k_m <= 0.95)
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * (k_m ^ 4.0))) / l));
	else
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / cos(k_m))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(1.0 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.08611111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.25e-139], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.2e-79], N[(N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.95], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t * N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.25000000000000008e-139

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 1.25000000000000008e-139 < k < 2.1999999999999999e-79

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 61.8%

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

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. 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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.9%

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

   9. Applied egg-rr 62.9%

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

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

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

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

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

       12. /-lowering-/.f64 92.4%

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

   11. Applied egg-rr 92.4%

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

   if 2.1999999999999999e-79 < k < 0.94999999999999996

   1. Initial program 59.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)}\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{{t}^{3}}{{\ell}^{2}}\right), \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

   5. Simplified 59.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(0.08611111111111111 \cdot \left(k \cdot k\right)\right) + \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right) + \color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right), \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)\right)}\right)\right)\right) \]

   8. Simplified 91.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)}} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{{\ell}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{\ell \cdot \color{blue}{\ell}}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t}{\ell} \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   11. Simplified 92.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + {k}^{4} \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)}{\ell}}} \]

   if 0.94999999999999996 < k

   1. Initial program 50.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

   9. Applied egg-rr 91.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(0.5 + \cos \left(2 \cdot k\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k}{\cos k}}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 0.95:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right) \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{2}{\left({t}^{1.5} \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k\_m \cdot \left(2 + \frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
       (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0))))
      INFINITY)
   (/
    2.0
    (*
     (* (pow t 1.5) (/ (sin k_m) l))
     (* (/ (pow t 1.5) l) (* (tan k_m) (+ 2.0 (/ (/ k_m t) (/ t k_m)))))))
   (/
    2.0
    (*
     (/ (* k_m (+ 0.5 (* (cos (* k_m 2.0)) -0.5))) l)
     (/ (* t (/ k_m (cos k_m))) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * (1.0 + (1.0 + pow((k_m / t), 2.0)))) <= ((double) INFINITY)) {
		tmp = 2.0 / ((pow(t, 1.5) * (sin(k_m) / l)) * ((pow(t, 1.5) / l) * (tan(k_m) * (2.0 + ((k_m / t) / (t / k_m))))));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / cos(k_m))) / l));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * (1.0 + (1.0 + Math.pow((k_m / t), 2.0)))) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / ((Math.pow(t, 1.5) * (Math.sin(k_m) / l)) * ((Math.pow(t, 1.5) / l) * (Math.tan(k_m) * (2.0 + ((k_m / t) / (t / k_m))))));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (Math.cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / Math.cos(k_m))) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * (1.0 + (1.0 + math.pow((k_m / t), 2.0)))) <= math.inf:
		tmp = 2.0 / ((math.pow(t, 1.5) * (math.sin(k_m) / l)) * ((math.pow(t, 1.5) / l) * (math.tan(k_m) * (2.0 + ((k_m / t) / (t / k_m))))))
	else:
		tmp = 2.0 / (((k_m * (0.5 + (math.cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / math.cos(k_m))) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0)))) <= Inf)
		tmp = Float64(2.0 / Float64(Float64((t ^ 1.5) * Float64(sin(k_m) / l)) * Float64(Float64((t ^ 1.5) / l) * Float64(tan(k_m) * Float64(2.0 + Float64(Float64(k_m / t) / Float64(t / k_m)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))) / l) * Float64(Float64(t * Float64(k_m / cos(k_m))) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * (1.0 + (1.0 + ((k_m / t) ^ 2.0)))) <= Inf)
		tmp = 2.0 / (((t ^ 1.5) * (sin(k_m) / l)) * (((t ^ 1.5) / l) * (tan(k_m) * (2.0 + ((k_m / t) / (t / k_m))))));
	else
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / cos(k_m))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 / N[(N[(N[Power[t, 1.5], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t * N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 81.0%

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

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

      4. sqr-pow N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

      15. sin-lowering-sin.f64 39.1%

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

   4. Applied egg-rr 39.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-commutative N/A

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

      14. associate-+r+ N/A

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

      15. metadata-eval N/A

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

      16. unpow2 N/A

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

      17. frac-times N/A

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

   6. Applied egg-rr 39.2%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 51.6%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 55.8%

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

   7. Applied egg-rr 55.8%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

   9. Applied egg-rr 76.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{2}{\left({t}^{1.5} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 1 + \left(k\_m \cdot k\_m\right) \cdot \left(0.16666666666666666 + \left(k\_m \cdot k\_m\right) \cdot 0.08611111111111111\right)\\ \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k\_m \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k\_m \cdot {t}^{1.5}}{\ell}\right)\right)}\\ \mathbf{elif}\;k\_m \leq 0.2:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot t\right)\right) \cdot t\_1\right) + t\_1 \cdot {k\_m}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1
         (+
          1.0
          (*
           (* k_m k_m)
           (+ 0.16666666666666666 (* (* k_m k_m) 0.08611111111111111))))))
   (if (<= k_m 1.3e-81)
     (/
      2.0
      (*
       2.0
       (* (tan k_m) (* (/ (pow t 1.5) l) (/ (* (sin k_m) (pow t 1.5)) l)))))
     (if (<= k_m 0.2)
       (/
        2.0
        (*
         (/ t l)
         (/
          (+ (* 2.0 (* (* (* k_m k_m) (* t t)) t_1)) (* t_1 (pow k_m 4.0)))
          l)))
       (/
        2.0
        (*
         (/ (* k_m (+ 0.5 (* (cos (* k_m 2.0)) -0.5))) l)
         (/ (* t (/ k_m (cos k_m))) l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 1.3e-81) {
		tmp = 2.0 / (2.0 * (tan(k_m) * ((pow(t, 1.5) / l) * ((sin(k_m) * pow(t, 1.5)) / l))));
	} else if (k_m <= 0.2) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * pow(k_m, 4.0))) / l));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / cos(k_m))) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + ((k_m * k_m) * (0.16666666666666666d0 + ((k_m * k_m) * 0.08611111111111111d0)))
    if (k_m <= 1.3d-81) then
        tmp = 2.0d0 / (2.0d0 * (tan(k_m) * (((t ** 1.5d0) / l) * ((sin(k_m) * (t ** 1.5d0)) / l))))
    else if (k_m <= 0.2d0) then
        tmp = 2.0d0 / ((t / l) * (((2.0d0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * (k_m ** 4.0d0))) / l))
    else
        tmp = 2.0d0 / (((k_m * (0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0)))) / l) * ((t * (k_m / cos(k_m))) / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 1.3e-81) {
		tmp = 2.0 / (2.0 * (Math.tan(k_m) * ((Math.pow(t, 1.5) / l) * ((Math.sin(k_m) * Math.pow(t, 1.5)) / l))));
	} else if (k_m <= 0.2) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * Math.pow(k_m, 4.0))) / l));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (Math.cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / Math.cos(k_m))) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)))
	tmp = 0
	if k_m <= 1.3e-81:
		tmp = 2.0 / (2.0 * (math.tan(k_m) * ((math.pow(t, 1.5) / l) * ((math.sin(k_m) * math.pow(t, 1.5)) / l))))
	elif k_m <= 0.2:
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * math.pow(k_m, 4.0))) / l))
	else:
		tmp = 2.0 / (((k_m * (0.5 + (math.cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / math.cos(k_m))) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(1.0 + Float64(Float64(k_m * k_m) * Float64(0.16666666666666666 + Float64(Float64(k_m * k_m) * 0.08611111111111111))))
	tmp = 0.0
	if (k_m <= 1.3e-81)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k_m) * Float64(Float64((t ^ 1.5) / l) * Float64(Float64(sin(k_m) * (t ^ 1.5)) / l)))));
	elseif (k_m <= 0.2)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(2.0 * Float64(Float64(Float64(k_m * k_m) * Float64(t * t)) * t_1)) + Float64(t_1 * (k_m ^ 4.0))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))) / l) * Float64(Float64(t * Float64(k_m / cos(k_m))) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	tmp = 0.0;
	if (k_m <= 1.3e-81)
		tmp = 2.0 / (2.0 * (tan(k_m) * (((t ^ 1.5) / l) * ((sin(k_m) * (t ^ 1.5)) / l))));
	elseif (k_m <= 0.2)
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_1)) + (t_1 * (k_m ^ 4.0))) / l));
	else
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * ((t * (k_m / cos(k_m))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(1.0 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.08611111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.3e-81], N[(2.0 / N[(2.0 * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.2], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t * N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.2999999999999999e-81

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. cube-unmult N/A

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

      4. sqr-pow N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

      15. sin-lowering-sin.f64 33.0%

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

   4. Applied egg-rr 33.0%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 30.1%

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

   if 1.2999999999999999e-81 < k < 0.20000000000000001

   1. Initial program 59.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)}\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{{t}^{3}}{{\ell}^{2}}\right), \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

   5. Simplified 59.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(0.08611111111111111 \cdot \left(k \cdot k\right)\right) + \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right) + \color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right), \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)\right)}\right)\right)\right) \]

   8. Simplified 91.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)}} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{{\ell}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{\ell \cdot \color{blue}{\ell}}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t}{\ell} \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   11. Simplified 92.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + {k}^{4} \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)}{\ell}}} \]

   if 0.20000000000000001 < k

   1. Initial program 50.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

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

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

   9. Applied egg-rr 91.3%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 0.2:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right) \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \frac{t \cdot \frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos \left(k\_m \cdot 2\right)\\ t_2 := t \cdot \left(k\_m \cdot k\_m\right)\\ t_3 := 1 + \left(k\_m \cdot k\_m\right) \cdot \left(0.16666666666666666 + \left(k\_m \cdot k\_m\right) \cdot 0.08611111111111111\right)\\ \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\ell}{t \cdot t\_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k\_m \leq 6.4:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot t\right)\right) \cdot t\_3\right) + t\_3 \cdot {k\_m}^{4}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(0.5 - 0.5 \cdot t\_1\right) \cdot \frac{t\_2}{\cos k\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\_m\right) \cdot \frac{k\_m \cdot \left(0.5 + t\_1 \cdot -0.5\right)}{\ell \cdot \left(\ell \cdot \cos k\_m\right)}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (cos (* k_m 2.0)))
        (t_2 (* t (* k_m k_m)))
        (t_3
         (+
          1.0
          (*
           (* k_m k_m)
           (+ 0.16666666666666666 (* (* k_m k_m) 0.08611111111111111))))))
   (if (<= k_m 1.5e-139)
     (* l (/ l (* (* t k_m) (* t (* t k_m)))))
     (if (<= k_m 8e-82)
       (* (/ l (* t t_2)) (/ l t))
       (if (<= k_m 6.4)
         (/
          2.0
          (*
           (/ t l)
           (/
            (+ (* 2.0 (* (* (* k_m k_m) (* t t)) t_3)) (* t_3 (pow k_m 4.0)))
            l)))
         (if (<= k_m 7e+152)
           (* l (* l (/ 2.0 (* (- 0.5 (* 0.5 t_1)) (/ t_2 (cos k_m))))))
           (/
            2.0
            (*
             (* t k_m)
             (/ (* k_m (+ 0.5 (* t_1 -0.5))) (* l (* l (cos k_m))))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos((k_m * 2.0));
	double t_2 = t * (k_m * k_m);
	double t_3 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 1.5e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 8e-82) {
		tmp = (l / (t * t_2)) * (l / t);
	} else if (k_m <= 6.4) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_3)) + (t_3 * pow(k_m, 4.0))) / l));
	} else if (k_m <= 7e+152) {
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * t_1)) * (t_2 / cos(k_m)))));
	} else {
		tmp = 2.0 / ((t * k_m) * ((k_m * (0.5 + (t_1 * -0.5))) / (l * (l * cos(k_m)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = cos((k_m * 2.0d0))
    t_2 = t * (k_m * k_m)
    t_3 = 1.0d0 + ((k_m * k_m) * (0.16666666666666666d0 + ((k_m * k_m) * 0.08611111111111111d0)))
    if (k_m <= 1.5d-139) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else if (k_m <= 8d-82) then
        tmp = (l / (t * t_2)) * (l / t)
    else if (k_m <= 6.4d0) then
        tmp = 2.0d0 / ((t / l) * (((2.0d0 * (((k_m * k_m) * (t * t)) * t_3)) + (t_3 * (k_m ** 4.0d0))) / l))
    else if (k_m <= 7d+152) then
        tmp = l * (l * (2.0d0 / ((0.5d0 - (0.5d0 * t_1)) * (t_2 / cos(k_m)))))
    else
        tmp = 2.0d0 / ((t * k_m) * ((k_m * (0.5d0 + (t_1 * (-0.5d0)))) / (l * (l * cos(k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos((k_m * 2.0));
	double t_2 = t * (k_m * k_m);
	double t_3 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 1.5e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 8e-82) {
		tmp = (l / (t * t_2)) * (l / t);
	} else if (k_m <= 6.4) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_3)) + (t_3 * Math.pow(k_m, 4.0))) / l));
	} else if (k_m <= 7e+152) {
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * t_1)) * (t_2 / Math.cos(k_m)))));
	} else {
		tmp = 2.0 / ((t * k_m) * ((k_m * (0.5 + (t_1 * -0.5))) / (l * (l * Math.cos(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos((k_m * 2.0))
	t_2 = t * (k_m * k_m)
	t_3 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)))
	tmp = 0
	if k_m <= 1.5e-139:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	elif k_m <= 8e-82:
		tmp = (l / (t * t_2)) * (l / t)
	elif k_m <= 6.4:
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_3)) + (t_3 * math.pow(k_m, 4.0))) / l))
	elif k_m <= 7e+152:
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * t_1)) * (t_2 / math.cos(k_m)))))
	else:
		tmp = 2.0 / ((t * k_m) * ((k_m * (0.5 + (t_1 * -0.5))) / (l * (l * math.cos(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cos(Float64(k_m * 2.0))
	t_2 = Float64(t * Float64(k_m * k_m))
	t_3 = Float64(1.0 + Float64(Float64(k_m * k_m) * Float64(0.16666666666666666 + Float64(Float64(k_m * k_m) * 0.08611111111111111))))
	tmp = 0.0
	if (k_m <= 1.5e-139)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	elseif (k_m <= 8e-82)
		tmp = Float64(Float64(l / Float64(t * t_2)) * Float64(l / t));
	elseif (k_m <= 6.4)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(2.0 * Float64(Float64(Float64(k_m * k_m) * Float64(t * t)) * t_3)) + Float64(t_3 * (k_m ^ 4.0))) / l)));
	elseif (k_m <= 7e+152)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * t_1)) * Float64(t_2 / cos(k_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * k_m) * Float64(Float64(k_m * Float64(0.5 + Float64(t_1 * -0.5))) / Float64(l * Float64(l * cos(k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos((k_m * 2.0));
	t_2 = t * (k_m * k_m);
	t_3 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	tmp = 0.0;
	if (k_m <= 1.5e-139)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	elseif (k_m <= 8e-82)
		tmp = (l / (t * t_2)) * (l / t);
	elseif (k_m <= 6.4)
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_3)) + (t_3 * (k_m ^ 4.0))) / l));
	elseif (k_m <= 7e+152)
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * t_1)) * (t_2 / cos(k_m)))));
	else
		tmp = 2.0 / ((t * k_m) * ((k_m * (0.5 + (t_1 * -0.5))) / (l * (l * cos(k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.08611111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.5e-139], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8e-82], N[(N[(l / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6.4], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7e+152], N[(l * N[(l * N[(2.0 / N[(N[(0.5 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * k$95$m), $MachinePrecision] * N[(N[(k$95$m * N[(0.5 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < 1.5e-139

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 1.5e-139 < k < 8e-82

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 61.8%

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

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. 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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.9%

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

   9. Applied egg-rr 62.9%

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

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

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

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

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

       12. /-lowering-/.f64 92.4%

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

   11. Applied egg-rr 92.4%

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

   if 8e-82 < k < 6.4000000000000004

   1. Initial program 59.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)}\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{{t}^{3}}{{\ell}^{2}}\right), \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

   5. Simplified 59.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(0.08611111111111111 \cdot \left(k \cdot k\right)\right) + \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right) + \color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right), \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)\right)}\right)\right)\right) \]

   8. Simplified 91.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)}} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{{\ell}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{\ell \cdot \color{blue}{\ell}}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t}{\ell} \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   11. Simplified 92.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + {k}^{4} \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)}{\ell}}} \]

   if 6.4000000000000004 < k < 6.99999999999999963e152

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 85.3%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

   7. Applied egg-rr 91.2%

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

   if 6.99999999999999963e152 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 61.3%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 68.5%

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

   7. Applied egg-rr 68.5%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 75.1%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot k\right) \cdot \frac{k \cdot \left(0.5 + \cos \left(2 \cdot k\right) \cdot -0.5\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 6.4:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right) \cdot {k}^{4}}{\ell}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\cos k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \frac{k \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \left(k\_m \cdot k\_m\right)\\ t_2 := 1 + \left(k\_m \cdot k\_m\right) \cdot \left(0.16666666666666666 + \left(k\_m \cdot k\_m\right) \cdot 0.08611111111111111\right)\\ \mathbf{if}\;k\_m \leq 3 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 10^{-83}:\\ \;\;\;\;\frac{\ell}{t \cdot t\_1} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k\_m \leq 0.47:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot t\right)\right) \cdot t\_2\right) + t\_2 \cdot {k\_m}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)\right) \cdot \frac{t\_1}{\cos k\_m}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (* k_m k_m)))
        (t_2
         (+
          1.0
          (*
           (* k_m k_m)
           (+ 0.16666666666666666 (* (* k_m k_m) 0.08611111111111111))))))
   (if (<= k_m 3e-139)
     (* l (/ l (* (* t k_m) (* t (* t k_m)))))
     (if (<= k_m 1e-83)
       (* (/ l (* t t_1)) (/ l t))
       (if (<= k_m 0.47)
         (/
          2.0
          (*
           (/ t l)
           (/
            (+ (* 2.0 (* (* (* k_m k_m) (* t t)) t_2)) (* t_2 (pow k_m 4.0)))
            l)))
         (*
          l
          (*
           l
           (/
            2.0
            (* (- 0.5 (* 0.5 (cos (* k_m 2.0)))) (/ t_1 (cos k_m)))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * k_m);
	double t_2 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 3e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 1e-83) {
		tmp = (l / (t * t_1)) * (l / t);
	} else if (k_m <= 0.47) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_2)) + (t_2 * pow(k_m, 4.0))) / l));
	} else {
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * cos((k_m * 2.0)))) * (t_1 / cos(k_m)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (k_m * k_m)
    t_2 = 1.0d0 + ((k_m * k_m) * (0.16666666666666666d0 + ((k_m * k_m) * 0.08611111111111111d0)))
    if (k_m <= 3d-139) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else if (k_m <= 1d-83) then
        tmp = (l / (t * t_1)) * (l / t)
    else if (k_m <= 0.47d0) then
        tmp = 2.0d0 / ((t / l) * (((2.0d0 * (((k_m * k_m) * (t * t)) * t_2)) + (t_2 * (k_m ** 4.0d0))) / l))
    else
        tmp = l * (l * (2.0d0 / ((0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))) * (t_1 / cos(k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * k_m);
	double t_2 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	double tmp;
	if (k_m <= 3e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 1e-83) {
		tmp = (l / (t * t_1)) * (l / t);
	} else if (k_m <= 0.47) {
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_2)) + (t_2 * Math.pow(k_m, 4.0))) / l));
	} else {
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * Math.cos((k_m * 2.0)))) * (t_1 / Math.cos(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = t * (k_m * k_m)
	t_2 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)))
	tmp = 0
	if k_m <= 3e-139:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	elif k_m <= 1e-83:
		tmp = (l / (t * t_1)) * (l / t)
	elif k_m <= 0.47:
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_2)) + (t_2 * math.pow(k_m, 4.0))) / l))
	else:
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * math.cos((k_m * 2.0)))) * (t_1 / math.cos(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * Float64(k_m * k_m))
	t_2 = Float64(1.0 + Float64(Float64(k_m * k_m) * Float64(0.16666666666666666 + Float64(Float64(k_m * k_m) * 0.08611111111111111))))
	tmp = 0.0
	if (k_m <= 3e-139)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	elseif (k_m <= 1e-83)
		tmp = Float64(Float64(l / Float64(t * t_1)) * Float64(l / t));
	elseif (k_m <= 0.47)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(2.0 * Float64(Float64(Float64(k_m * k_m) * Float64(t * t)) * t_2)) + Float64(t_2 * (k_m ^ 4.0))) / l)));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0)))) * Float64(t_1 / cos(k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = t * (k_m * k_m);
	t_2 = 1.0 + ((k_m * k_m) * (0.16666666666666666 + ((k_m * k_m) * 0.08611111111111111)));
	tmp = 0.0;
	if (k_m <= 3e-139)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	elseif (k_m <= 1e-83)
		tmp = (l / (t * t_1)) * (l / t);
	elseif (k_m <= 0.47)
		tmp = 2.0 / ((t / l) * (((2.0 * (((k_m * k_m) * (t * t)) * t_2)) + (t_2 * (k_m ^ 4.0))) / l));
	else
		tmp = l * (l * (2.0 / ((0.5 - (0.5 * cos((k_m * 2.0)))) * (t_1 / cos(k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.08611111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 3e-139], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1e-83], N[(N[(l / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.47], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 2.9999999999999999e-139

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 2.9999999999999999e-139 < k < 1e-83

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 61.8%

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

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. 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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.9%

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

   9. Applied egg-rr 62.9%

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

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

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

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

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

       12. /-lowering-/.f64 92.4%

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

   11. Applied egg-rr 92.4%

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

   if 1e-83 < k < 0.46999999999999997

   1. Initial program 59.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)}\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{{t}^{3}}{{\ell}^{2}}\right), \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

   5. Simplified 59.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(0.08611111111111111 \cdot \left(k \cdot k\right)\right) + \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right) + \color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right), \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)\right)}\right)\right)\right) \]

   8. Simplified 91.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)}} \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{{\ell}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}{\ell \cdot \color{blue}{\ell}}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t}{\ell} \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}{\ell}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}}{\ell}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)\right) + {k}^{4} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   11. Simplified 92.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + {k}^{4} \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)}{\ell}}} \]

   if 0.46999999999999997 < k

   1. Initial program 50.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 74.3%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

   7. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\cos k}} \cdot \ell\right) \cdot \ell} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 10^{-83}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 0.47:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right)\right) + \left(1 + \left(k \cdot k\right) \cdot \left(0.16666666666666666 + \left(k \cdot k\right) \cdot 0.08611111111111111\right)\right) \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\cos k}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \left(k\_m \cdot k\_m\right)\\ \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 2.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\ell}{t \cdot t\_1} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k\_m \leq 1.95 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k\_m \cdot k\_m\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \frac{\left(t \cdot t\right) \cdot 0.3333333333333333}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (* k_m k_m))))
   (if (<= k_m 1.2e-139)
     (* l (/ l (* (* t k_m) (* t (* t k_m)))))
     (if (<= k_m 2.8e-79)
       (* (/ l (* t t_1)) (/ l t))
       (if (<= k_m 1.95e+68)
         (/
          2.0
          (*
           t
           (*
            (* k_m k_m)
            (+
             (/ (* 2.0 (* t t)) (* l l))
             (*
              (* k_m k_m)
              (+
               (/ 1.0 (* l l))
               (/ (* (* t t) 0.3333333333333333) (* l l))))))))
         (/ (* (* l l) -0.3333333333333333) t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * k_m);
	double tmp;
	if (k_m <= 1.2e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 2.8e-79) {
		tmp = (l / (t * t_1)) * (l / t);
	} else if (k_m <= 1.95e+68) {
		tmp = 2.0 / (t * ((k_m * k_m) * (((2.0 * (t * t)) / (l * l)) + ((k_m * k_m) * ((1.0 / (l * l)) + (((t * t) * 0.3333333333333333) / (l * l)))))));
	} else {
		tmp = ((l * l) * -0.3333333333333333) / t_1;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (k_m * k_m)
    if (k_m <= 1.2d-139) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else if (k_m <= 2.8d-79) then
        tmp = (l / (t * t_1)) * (l / t)
    else if (k_m <= 1.95d+68) then
        tmp = 2.0d0 / (t * ((k_m * k_m) * (((2.0d0 * (t * t)) / (l * l)) + ((k_m * k_m) * ((1.0d0 / (l * l)) + (((t * t) * 0.3333333333333333d0) / (l * l)))))))
    else
        tmp = ((l * l) * (-0.3333333333333333d0)) / t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * k_m);
	double tmp;
	if (k_m <= 1.2e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 2.8e-79) {
		tmp = (l / (t * t_1)) * (l / t);
	} else if (k_m <= 1.95e+68) {
		tmp = 2.0 / (t * ((k_m * k_m) * (((2.0 * (t * t)) / (l * l)) + ((k_m * k_m) * ((1.0 / (l * l)) + (((t * t) * 0.3333333333333333) / (l * l)))))));
	} else {
		tmp = ((l * l) * -0.3333333333333333) / t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = t * (k_m * k_m)
	tmp = 0
	if k_m <= 1.2e-139:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	elif k_m <= 2.8e-79:
		tmp = (l / (t * t_1)) * (l / t)
	elif k_m <= 1.95e+68:
		tmp = 2.0 / (t * ((k_m * k_m) * (((2.0 * (t * t)) / (l * l)) + ((k_m * k_m) * ((1.0 / (l * l)) + (((t * t) * 0.3333333333333333) / (l * l)))))))
	else:
		tmp = ((l * l) * -0.3333333333333333) / t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 1.2e-139)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	elseif (k_m <= 2.8e-79)
		tmp = Float64(Float64(l / Float64(t * t_1)) * Float64(l / t));
	elseif (k_m <= 1.95e+68)
		tmp = Float64(2.0 / Float64(t * Float64(Float64(k_m * k_m) * Float64(Float64(Float64(2.0 * Float64(t * t)) / Float64(l * l)) + Float64(Float64(k_m * k_m) * Float64(Float64(1.0 / Float64(l * l)) + Float64(Float64(Float64(t * t) * 0.3333333333333333) / Float64(l * l))))))));
	else
		tmp = Float64(Float64(Float64(l * l) * -0.3333333333333333) / t_1);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = t * (k_m * k_m);
	tmp = 0.0;
	if (k_m <= 1.2e-139)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	elseif (k_m <= 2.8e-79)
		tmp = (l / (t * t_1)) * (l / t);
	elseif (k_m <= 1.95e+68)
		tmp = 2.0 / (t * ((k_m * k_m) * (((2.0 * (t * t)) / (l * l)) + ((k_m * k_m) * ((1.0 / (l * l)) + (((t * t) * 0.3333333333333333) / (l * l)))))));
	else
		tmp = ((l * l) * -0.3333333333333333) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.2e-139], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.8e-79], N[(N[(l / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.95e+68], N[(2.0 / N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.20000000000000007e-139

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 1.20000000000000007e-139 < k < 2.80000000000000012e-79

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 61.8%

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

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. 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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.9%

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

   9. Applied egg-rr 62.9%

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

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

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

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

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

       12. /-lowering-/.f64 92.4%

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

   11. Applied egg-rr 92.4%

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

   if 2.80000000000000012e-79 < k < 1.95000000000000009e68

   1. Initial program 52.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)}\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{{t}^{3}}{{\ell}^{2}}\right), \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

   5. Simplified 49.0%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(0.08611111111111111 \cdot \left(k \cdot k\right)\right) + \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right) + \color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right), \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)\right)}\right)\right)\right) \]

   8. Simplified 68.9%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)}} \]

   9. Taylor expanded in k around 0

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

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

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

       2. unpow2 N/A

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

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

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

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

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

       5. associate-*r/ N/A

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

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

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

       13. unpow2 N/A

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

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

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

       15. +-commutative N/A

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

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

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

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

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

       18. unpow2 N/A

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

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

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

       20. associate-*r/ N/A

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

   11. Simplified 69.2%

       \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \frac{0.3333333333333333 \cdot \left(t \cdot t\right)}{\ell \cdot \ell}\right)\right)\right)}} \]

   if 1.95000000000000009e68 < k

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 71.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) + \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 62.3%

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

   11. Simplified 62.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \frac{\left(t \cdot t\right) \cdot 0.3333333333333333}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{1}{\ell \cdot \ell}\\ \mathbf{if}\;k\_m \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t\_1 + \left(k\_m \cdot k\_m\right) \cdot \left(t\_1 \cdot \left(0.16666666666666666 + k\_m \cdot \left(k\_m \cdot 0.08611111111111111\right)\right)\right)\right) \cdot \left(k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) + 2 \cdot \left(t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ 1.0 (* l l))))
   (if (<= k_m 4.5e-135)
     (* l (/ l (* (* t k_m) (* t (* t k_m)))))
     (/
      (/ 2.0 t)
      (*
       (+
        t_1
        (*
         (* k_m k_m)
         (* t_1 (+ 0.16666666666666666 (* k_m (* k_m 0.08611111111111111))))))
       (+ (* k_m (* k_m (* k_m k_m))) (* 2.0 (* t (* t (* k_m k_m))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 1.0 / (l * l);
	double tmp;
	if (k_m <= 4.5e-135) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (2.0 / t) / ((t_1 + ((k_m * k_m) * (t_1 * (0.16666666666666666 + (k_m * (k_m * 0.08611111111111111)))))) * ((k_m * (k_m * (k_m * k_m))) + (2.0 * (t * (t * (k_m * k_m))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / (l * l)
    if (k_m <= 4.5d-135) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else
        tmp = (2.0d0 / t) / ((t_1 + ((k_m * k_m) * (t_1 * (0.16666666666666666d0 + (k_m * (k_m * 0.08611111111111111d0)))))) * ((k_m * (k_m * (k_m * k_m))) + (2.0d0 * (t * (t * (k_m * k_m))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 1.0 / (l * l);
	double tmp;
	if (k_m <= 4.5e-135) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (2.0 / t) / ((t_1 + ((k_m * k_m) * (t_1 * (0.16666666666666666 + (k_m * (k_m * 0.08611111111111111)))))) * ((k_m * (k_m * (k_m * k_m))) + (2.0 * (t * (t * (k_m * k_m))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 1.0 / (l * l)
	tmp = 0
	if k_m <= 4.5e-135:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	else:
		tmp = (2.0 / t) / ((t_1 + ((k_m * k_m) * (t_1 * (0.16666666666666666 + (k_m * (k_m * 0.08611111111111111)))))) * ((k_m * (k_m * (k_m * k_m))) + (2.0 * (t * (t * (k_m * k_m))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(1.0 / Float64(l * l))
	tmp = 0.0
	if (k_m <= 4.5e-135)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	else
		tmp = Float64(Float64(2.0 / t) / Float64(Float64(t_1 + Float64(Float64(k_m * k_m) * Float64(t_1 * Float64(0.16666666666666666 + Float64(k_m * Float64(k_m * 0.08611111111111111)))))) * Float64(Float64(k_m * Float64(k_m * Float64(k_m * k_m))) + Float64(2.0 * Float64(t * Float64(t * Float64(k_m * k_m)))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 1.0 / (l * l);
	tmp = 0.0;
	if (k_m <= 4.5e-135)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	else
		tmp = (2.0 / t) / ((t_1 + ((k_m * k_m) * (t_1 * (0.16666666666666666 + (k_m * (k_m * 0.08611111111111111)))))) * ((k_m * (k_m * (k_m * k_m))) + (2.0 * (t * (t * (k_m * k_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 4.5e-135], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] / N[(N[(t$95$1 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$1 * N[(0.16666666666666666 + N[(k$95$m * N[(k$95$m * 0.08611111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.49999999999999987e-135

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 4.49999999999999987e-135 < k

   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 k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right)}, 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) + \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)}\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{{t}^{3}}{{\ell}^{2}}\right), \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), \color{blue}{2}\right)\right), 1\right)\right)\right) \]

   5. Simplified 51.5%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(0.08611111111111111 \cdot \left(k \cdot k\right)\right) + \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(t \cdot \left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right) + {k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right) + \color{blue}{2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right), \color{blue}{\left(2 \cdot \left({k}^{2} \cdot \left({t}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right)\right)\right)}\right)\right)\right) \]

   8. Simplified 57.2%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111 \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{0.16666666666666666}{\ell \cdot \ell}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{{k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right) + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{{k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right)} + 2 \cdot \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left({k}^{4} \cdot \left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right) + \left(2 \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{\frac{31}{360} \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + \frac{\frac{1}{6}}{\ell \cdot \ell}\right)\right)}\right)\right) \]

   10. Applied egg-rr 64.8%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot 0.08611111111111111\right) + 0.16666666666666666\right)\right)\right) \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right) + 2 \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(\frac{1}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(0.16666666666666666 + k \cdot \left(k \cdot 0.08611111111111111\right)\right)\right)\right) \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right) + 2 \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \left(k\_m \cdot k\_m\right)\\ \mathbf{if}\;k\_m \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+33}:\\ \;\;\;\;\frac{\ell}{t \cdot t\_1} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (* k_m k_m))))
   (if (<= k_m 2e-139)
     (* l (/ l (* (* t k_m) (* t (* t k_m)))))
     (if (<= k_m 3e+33)
       (* (/ l (* t t_1)) (/ l t))
       (/ (* (* l l) -0.3333333333333333) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * k_m);
	double tmp;
	if (k_m <= 2e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 3e+33) {
		tmp = (l / (t * t_1)) * (l / t);
	} else {
		tmp = ((l * l) * -0.3333333333333333) / t_1;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (k_m * k_m)
    if (k_m <= 2d-139) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else if (k_m <= 3d+33) then
        tmp = (l / (t * t_1)) * (l / t)
    else
        tmp = ((l * l) * (-0.3333333333333333d0)) / t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * k_m);
	double tmp;
	if (k_m <= 2e-139) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else if (k_m <= 3e+33) {
		tmp = (l / (t * t_1)) * (l / t);
	} else {
		tmp = ((l * l) * -0.3333333333333333) / t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = t * (k_m * k_m)
	tmp = 0
	if k_m <= 2e-139:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	elif k_m <= 3e+33:
		tmp = (l / (t * t_1)) * (l / t)
	else:
		tmp = ((l * l) * -0.3333333333333333) / t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 2e-139)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	elseif (k_m <= 3e+33)
		tmp = Float64(Float64(l / Float64(t * t_1)) * Float64(l / t));
	else
		tmp = Float64(Float64(Float64(l * l) * -0.3333333333333333) / t_1);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = t * (k_m * k_m);
	tmp = 0.0;
	if (k_m <= 2e-139)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	elseif (k_m <= 3e+33)
		tmp = (l / (t * t_1)) * (l / t);
	else
		tmp = ((l * l) * -0.3333333333333333) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2e-139], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+33], N[(N[(l / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.00000000000000006e-139

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 2.00000000000000006e-139 < k < 2.99999999999999984e33

   1. Initial program 56.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.0%

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

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 54.6%

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

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 54.9%

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

   9. Applied egg-rr 54.9%

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

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

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

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

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

       12. /-lowering-/.f64 71.2%

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

   11. Applied egg-rr 71.2%

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

   if 2.99999999999999984e33 < k

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

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

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

   5. Simplified 73.7%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   8. Simplified 29.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) + \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 63.3%

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

   11. Simplified 63.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-139}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+33}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.4% accurate, 23.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.6 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.6e-135)
   (* l (/ l (* (* t k_m) (* t (* t k_m)))))
   (/ (* l (/ l (* t (* t (* k_m k_m))))) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.6e-135) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l * (l / (t * (t * (k_m * k_m))))) / t;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.6d-135) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else
        tmp = (l * (l / (t * (t * (k_m * k_m))))) / t
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.6e-135) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l * (l / (t * (t * (k_m * k_m))))) / t;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.6e-135:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	else:
		tmp = (l * (l / (t * (t * (k_m * k_m))))) / t
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.6e-135)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	else
		tmp = Float64(Float64(l * Float64(l / Float64(t * Float64(t * Float64(k_m * k_m))))) / t);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.6e-135)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	else
		tmp = (l * (l / (t * (t * (k_m * k_m))))) / t;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.6e-135], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.60000000000000004e-135

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-lowering-*.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 2.60000000000000004e-135 < k

   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 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 51.9%

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

   5. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. 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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 56.2%

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

   9. Applied egg-rr 56.2%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

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

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

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

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

       13. *-lowering-*.f64 63.8%

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

   11. Applied egg-rr 63.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4e-135)
   (* l (/ l (* (* t k_m) (* t (* t k_m)))))
   (/ (* l (/ l t)) (* t (* t (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-135) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l * (l / t)) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4d-135) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else
        tmp = (l * (l / t)) / (t * (t * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-135) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = (l * (l / t)) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4e-135:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	else:
		tmp = (l * (l / t)) / (t * (t * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-135)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	else
		tmp = Float64(Float64(l * Float64(l / t)) / Float64(t * Float64(t * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4e-135)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	else
		tmp = (l * (l / t)) / (t * (t * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4e-135], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.0000000000000002e-135

   1. Initial program 59.5%

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

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

   5. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right), t\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right), t\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

      13. *-lowering-*.f64 62.7%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), t\right)\right)\right) \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right) \cdot t\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), \left(\color{blue}{t} \cdot k\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), \left(\color{blue}{t} \cdot k\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot k\right)\right)\right)\right) \]

       9. *-lowering-*.f64 71.7%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right)\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \]

   if 4.0000000000000002e-135 < k

   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 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 51.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 53.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \ell\right), \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \ell\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot t\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \left(\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot t\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{t}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \left(t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   9. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 23.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{+33}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\_m\right) \cdot \left(t \cdot \left(t \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e+33)
   (* l (/ l (* (* t k_m) (* t (* t k_m)))))
   (/ (* (* l l) -0.3333333333333333) (* t (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e+33) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = ((l * l) * -0.3333333333333333) / (t * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3d+33) then
        tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
    else
        tmp = ((l * l) * (-0.3333333333333333d0)) / (t * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e+33) {
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	} else {
		tmp = ((l * l) * -0.3333333333333333) / (t * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3e+33:
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))))
	else:
		tmp = ((l * l) * -0.3333333333333333) / (t * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e+33)
		tmp = Float64(l * Float64(l / Float64(Float64(t * k_m) * Float64(t * Float64(t * k_m)))));
	else
		tmp = Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(t * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e+33)
		tmp = l * (l / ((t * k_m) * (t * (t * k_m))));
	else
		tmp = ((l * l) * -0.3333333333333333) / (t * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e+33], N[(l * N[(l / N[(N[(t * k$95$m), $MachinePrecision] * N[(t * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999984e33

   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 59.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   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 55.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right), \color{blue}{t}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right), t\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right), t\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right), t\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

      13. *-lowering-*.f64 61.4%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), t\right)\right)\right) \]

   9. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right) \cdot t\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), \left(\color{blue}{t} \cdot k\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), \left(\color{blue}{t} \cdot k\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot k\right)\right)\right)\right) \]

       9. *-lowering-*.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right)\right) \]

   11. Applied egg-rr 68.9%

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \]

   if 2.99999999999999984e33 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}{\color{blue}{{\ell}^{2}}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\cos k}}{{\ell}^{2}}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}{\cos k}}{{\ell}^{2}}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)}{\cos k}}{{\ell}^{2}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\color{blue}{\ell}}^{2}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

   5. Simplified 73.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\cos k}}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   8. Simplified 29.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) + \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right) \]

       8. *-lowering-*.f64 63.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right) \]

   11. Simplified 63.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+33}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.9% accurate, 38.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* l l) -0.3333333333333333) (* t (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / (t * (k_m * k_m));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) * (-0.3333333333333333d0)) / (t * (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / (t * (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * -0.3333333333333333) / (t * (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(t * Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * -0.3333333333333333) / (t * (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}{\color{blue}{{\ell}^{2}}}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\cos k}}{{\ell}^{2}}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}{\cos k}}{{\ell}^{2}}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)}{\cos k}}{{\ell}^{2}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}}{{\color{blue}{\ell}}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

5. Simplified 58.9%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\cos k}}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{4}\right)}\right) \]

8. Simplified 22.7%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(k \cdot k\right) \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) + \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}}} \]

9. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right) \]

    8. *-lowering-*.f64 30.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right) \]

11. Simplified 30.2%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t}} \]

12. Final simplification 30.2%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t \cdot \left(k \cdot k\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))