Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 90.7%

Time: 19.1s

Alternatives: 19

Speedup: 32.4×

• 27.4% 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: 54.5% 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: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 7.6 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_2}}{t\_m}}{t\_m \cdot \left(\left(2 + \frac{k \cdot \frac{k}{t\_m}}{t\_m}\right) \cdot \left(t\_m \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m}}{k} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 1.5e-101)
      (/ 2.0 (* t_2 (/ (* k (* k (* t_m (sin k)))) (* l (cos k)))))
      (if (<= t_m 7.6e+192)
        (/
         (/ (/ 2.0 t_2) t_m)
         (* t_m (* (+ 2.0 (/ (* k (/ k t_m)) t_m)) (* t_m (/ (tan k) l)))))
        (* (/ (/ (/ l (* t_m k)) t_m) k) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 1.5e-101) {
		tmp = 2.0 / (t_2 * ((k * (k * (t_m * sin(k)))) / (l * cos(k))));
	} else if (t_m <= 7.6e+192) {
		tmp = ((2.0 / t_2) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (tan(k) / l))));
	} else {
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) / l
    if (t_m <= 1.5d-101) then
        tmp = 2.0d0 / (t_2 * ((k * (k * (t_m * sin(k)))) / (l * cos(k))))
    else if (t_m <= 7.6d+192) then
        tmp = ((2.0d0 / t_2) / t_m) / (t_m * ((2.0d0 + ((k * (k / t_m)) / t_m)) * (t_m * (tan(k) / l))))
    else
        tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) / l;
	double tmp;
	if (t_m <= 1.5e-101) {
		tmp = 2.0 / (t_2 * ((k * (k * (t_m * Math.sin(k)))) / (l * Math.cos(k))));
	} else if (t_m <= 7.6e+192) {
		tmp = ((2.0 / t_2) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (Math.tan(k) / l))));
	} else {
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sin(k) / l
	tmp = 0
	if t_m <= 1.5e-101:
		tmp = 2.0 / (t_2 * ((k * (k * (t_m * math.sin(k)))) / (l * math.cos(k))))
	elif t_m <= 7.6e+192:
		tmp = ((2.0 / t_2) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (math.tan(k) / l))))
	else:
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 1.5e-101)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k * Float64(k * Float64(t_m * sin(k)))) / Float64(l * cos(k)))));
	elseif (t_m <= 7.6e+192)
		tmp = Float64(Float64(Float64(2.0 / t_2) / t_m) / Float64(t_m * Float64(Float64(2.0 + Float64(Float64(k * Float64(k / t_m)) / t_m)) * Float64(t_m * Float64(tan(k) / l)))));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(t_m * k)) / t_m) / k) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) / l;
	tmp = 0.0;
	if (t_m <= 1.5e-101)
		tmp = 2.0 / (t_2 * ((k * (k * (t_m * sin(k)))) / (l * cos(k))));
	elseif (t_m <= 7.6e+192)
		tmp = ((2.0 / t_2) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (tan(k) / l))));
	else
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-101], N[(2.0 / N[(t$95$2 * N[(N[(k * N[(k * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.6e+192], N[(N[(N[(2.0 / t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(N[(2.0 + N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.5000000000000002e-101

   1. Initial program 60.6%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 58.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 66.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

   8. Applied egg-rr 77.9%

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

   9. Taylor expanded in t around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. cos-lowering-cos.f64 76.9%

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

   11. Simplified 76.9%

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

   if 1.5000000000000002e-101 < t < 7.5999999999999999e192

   1. Initial program 59.7%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 52.9%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 69.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

   8. Applied egg-rr 84.1%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

   10. Applied egg-rr 87.2%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 91.8%

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

   12. Applied egg-rr 91.8%

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

   if 7.5999999999999999e192 < t

   1. Initial program 47.5%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 36.7%

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

   5. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 37.0%

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

   7. Simplified 37.0%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. /-lowering-/.f64 57.8%

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

   9. Applied egg-rr 57.8%

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

       1. associate-*r* N/A

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

       2. unswap-sqr N/A

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

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

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

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

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

       5. *-lowering-*.f64 87.4%

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

   11. Applied egg-rr 87.4%

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

       1. associate-/r* N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 90.7%

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

   13. Applied egg-rr 90.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{t \cdot \left(\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right) \cdot \left(t \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t\_m}}{k}}{\sin k \cdot \left(k \cdot \tan k\right)}\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+193}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t\_m}}{t\_m \cdot \left(\left(2 + \frac{k \cdot \frac{k}{t\_m}}{t\_m}\right) \cdot \left(t\_m \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m}}{k} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e-52)
    (* l (* l (/ (/ (/ 2.0 t_m) k) (* (sin k) (* k (tan k))))))
    (if (<= t_m 5.6e+193)
      (/
       (/ (/ 2.0 (/ (sin k) l)) t_m)
       (* t_m (* (+ 2.0 (/ (* k (/ k t_m)) t_m)) (* t_m (/ (tan k) l)))))
      (* (/ (/ (/ l (* t_m k)) t_m) k) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-52) {
		tmp = l * (l * (((2.0 / t_m) / k) / (sin(k) * (k * tan(k)))));
	} else if (t_m <= 5.6e+193) {
		tmp = ((2.0 / (sin(k) / l)) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (tan(k) / l))));
	} else {
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.1d-52) then
        tmp = l * (l * (((2.0d0 / t_m) / k) / (sin(k) * (k * tan(k)))))
    else if (t_m <= 5.6d+193) then
        tmp = ((2.0d0 / (sin(k) / l)) / t_m) / (t_m * ((2.0d0 + ((k * (k / t_m)) / t_m)) * (t_m * (tan(k) / l))))
    else
        tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-52) {
		tmp = l * (l * (((2.0 / t_m) / k) / (Math.sin(k) * (k * Math.tan(k)))));
	} else if (t_m <= 5.6e+193) {
		tmp = ((2.0 / (Math.sin(k) / l)) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (Math.tan(k) / l))));
	} else {
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.1e-52:
		tmp = l * (l * (((2.0 / t_m) / k) / (math.sin(k) * (k * math.tan(k)))))
	elif t_m <= 5.6e+193:
		tmp = ((2.0 / (math.sin(k) / l)) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (math.tan(k) / l))))
	else:
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e-52)
		tmp = Float64(l * Float64(l * Float64(Float64(Float64(2.0 / t_m) / k) / Float64(sin(k) * Float64(k * tan(k))))));
	elseif (t_m <= 5.6e+193)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) / l)) / t_m) / Float64(t_m * Float64(Float64(2.0 + Float64(Float64(k * Float64(k / t_m)) / t_m)) * Float64(t_m * Float64(tan(k) / l)))));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(t_m * k)) / t_m) / k) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.1e-52)
		tmp = l * (l * (((2.0 / t_m) / k) / (sin(k) * (k * tan(k)))));
	elseif (t_m <= 5.6e+193)
		tmp = ((2.0 / (sin(k) / l)) / t_m) / (t_m * ((2.0 + ((k * (k / t_m)) / t_m)) * (t_m * (tan(k) / l))));
	else
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e-52], N[(l * N[(l * N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+193], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(N[(2.0 + N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.0999999999999999e-52

   1. Initial program 60.8%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 58.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 30.8%

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

   7. Simplified 30.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

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

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

   9. Applied egg-rr 32.6%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   11. Applied egg-rr 74.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\frac{\frac{1}{\tan k}}{\sin k}}}{\ell}} \cdot \ell} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

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

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

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

   13. Applied egg-rr 77.1%

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

   if 2.0999999999999999e-52 < t < 5.59999999999999972e193

   1. Initial program 58.7%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 50.4%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 68.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

   8. Applied egg-rr 86.3%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

   10. Applied egg-rr 90.2%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 95.8%

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

   12. Applied egg-rr 95.8%

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

   if 5.59999999999999972e193 < t

   1. Initial program 47.5%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 36.7%

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

   5. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 37.0%

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

   7. Simplified 37.0%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. /-lowering-/.f64 57.8%

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

   9. Applied egg-rr 57.8%

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

       1. associate-*r* N/A

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

       2. unswap-sqr N/A

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

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

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

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

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

       5. *-lowering-*.f64 87.4%

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

   11. Applied egg-rr 87.4%

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

       1. associate-/r* N/A

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

       2. associate-/r* N/A

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

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

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 90.7%

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

   13. Applied egg-rr 90.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t}}{k}}{\sin k \cdot \left(k \cdot \tan k\right)}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+193}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{t \cdot \left(\left(2 + \frac{k \cdot \frac{k}{t}}{t}\right) \cdot \left(t \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t\_m}}{k}}{\sin k \cdot \left(k \cdot \tan k\right)}\right)\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-52)
    (* l (* l (/ (/ (/ 2.0 t_m) k) (* (sin k) (* k (tan k))))))
    (if (<= t_m 5e+152)
      (/
       2.0
       (*
        (* t_m (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l))
        (* (/ (sin k) l) (* t_m t_m))))
      (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-52) {
		tmp = l * (l * (((2.0 / t_m) / k) / (sin(k) * (k * tan(k)))));
	} else if (t_m <= 5e+152) {
		tmp = 2.0 / ((t_m * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)) * ((sin(k) / l) * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.4d-52) then
        tmp = l * (l * (((2.0d0 / t_m) / k) / (sin(k) * (k * tan(k)))))
    else if (t_m <= 5d+152) then
        tmp = 2.0d0 / ((t_m * ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l)) * ((sin(k) / l) * (t_m * t_m)))
    else
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-52) {
		tmp = l * (l * (((2.0 / t_m) / k) / (Math.sin(k) * (k * Math.tan(k)))));
	} else if (t_m <= 5e+152) {
		tmp = 2.0 / ((t_m * ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)) * ((Math.sin(k) / l) * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.4e-52:
		tmp = l * (l * (((2.0 / t_m) / k) / (math.sin(k) * (k * math.tan(k)))))
	elif t_m <= 5e+152:
		tmp = 2.0 / ((t_m * ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)) * ((math.sin(k) / l) * (t_m * t_m)))
	else:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.4e-52)
		tmp = Float64(l * Float64(l * Float64(Float64(Float64(2.0 / t_m) / k) / Float64(sin(k) * Float64(k * tan(k))))));
	elseif (t_m <= 5e+152)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l)) * Float64(Float64(sin(k) / l) * Float64(t_m * t_m))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.4e-52)
		tmp = l * (l * (((2.0 / t_m) / k) / (sin(k) * (k * tan(k)))));
	elseif (t_m <= 5e+152)
		tmp = 2.0 / ((t_m * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)) * ((sin(k) / l) * (t_m * t_m)));
	else
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-52], N[(l * N[(l * N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+152], N[(2.0 / N[(N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.4000000000000002e-52

   1. Initial program 60.8%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 58.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 30.8%

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

   7. Simplified 30.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

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

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

   9. Applied egg-rr 32.6%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   11. Applied egg-rr 74.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\frac{\frac{1}{\tan k}}{\sin k}}}{\ell}} \cdot \ell} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

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

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

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

   13. Applied egg-rr 77.1%

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

   if 2.4000000000000002e-52 < t < 5e152

   1. Initial program 59.8%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 50.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 70.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

   8. Applied egg-rr 95.1%

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

   if 5e152 < t

   1. Initial program 48.1%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 38.9%

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

   5. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 39.3%

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

   7. Simplified 39.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. /-lowering-/.f64 59.5%

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

   9. Applied egg-rr 59.5%

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

       1. associate-/r* N/A

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

       2. associate-*r* N/A

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

       3. associate-/r* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 86.9%

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

   11. Applied egg-rr 86.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t}}{k}}{\sin k \cdot \left(k \cdot \tan k\right)}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{\left(t\_m \cdot k\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 4:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t\_m}}{k \cdot \left(\frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.6666666666666666\right)\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t\_m}}{k}}{\sin k \cdot \left(k \cdot \tan k\right)}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.7e-53)
    (/ (/ l (* t_m k)) (* (* t_m k) (/ t_m l)))
    (if (<= k 4.0)
      (/
       (/ (/ 2.0 (/ (sin k) l)) t_m)
       (*
        k
        (+
         (/ (* 2.0 (* t_m t_m)) l)
         (/ (* k (* k (+ 1.0 (* (* t_m t_m) 0.6666666666666666)))) l))))
      (* l (* l (/ (/ (/ 2.0 t_m) k) (* (sin k) (* k (tan k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.7e-53) {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	} else if (k <= 4.0) {
		tmp = ((2.0 / (sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)));
	} else {
		tmp = l * (l * (((2.0 / t_m) / k) / (sin(k) * (k * tan(k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.7d-53) then
        tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
    else if (k <= 4.0d0) then
        tmp = ((2.0d0 / (sin(k) / l)) / t_m) / (k * (((2.0d0 * (t_m * t_m)) / l) + ((k * (k * (1.0d0 + ((t_m * t_m) * 0.6666666666666666d0)))) / l)))
    else
        tmp = l * (l * (((2.0d0 / t_m) / k) / (sin(k) * (k * tan(k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.7e-53) {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	} else if (k <= 4.0) {
		tmp = ((2.0 / (Math.sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)));
	} else {
		tmp = l * (l * (((2.0 / t_m) / k) / (Math.sin(k) * (k * Math.tan(k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4.7e-53:
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
	elif k <= 4.0:
		tmp = ((2.0 / (math.sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)))
	else:
		tmp = l * (l * (((2.0 / t_m) / k) / (math.sin(k) * (k * math.tan(k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.7e-53)
		tmp = Float64(Float64(l / Float64(t_m * k)) / Float64(Float64(t_m * k) * Float64(t_m / l)));
	elseif (k <= 4.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) / l)) / t_m) / Float64(k * Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) / l) + Float64(Float64(k * Float64(k * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.6666666666666666)))) / l))));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(Float64(2.0 / t_m) / k) / Float64(sin(k) * Float64(k * tan(k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4.7e-53)
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	elseif (k <= 4.0)
		tmp = ((2.0 / (sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)));
	else
		tmp = l * (l * (((2.0 / t_m) / k) / (sin(k) * (k * tan(k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.7e-53], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.0], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(k * N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + N[(N[(k * N[(k * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.7e-53

   1. Initial program 57.7%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 51.3%

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

   5. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 56.1%

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

   7. Simplified 56.1%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. /-lowering-/.f64 66.8%

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

   9. Applied egg-rr 66.8%

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

       1. associate-*r* N/A

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

       2. unswap-sqr N/A

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

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

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

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

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

       5. *-lowering-*.f64 74.9%

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

   11. Applied egg-rr 74.9%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. associate-/r* N/A

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

       4. frac-times N/A

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

       5. clear-num N/A

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

       6. div-inv N/A

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

       7. clear-num N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 81.1%

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

   13. Applied egg-rr 81.1%

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

   if 4.7e-53 < k < 4

   1. Initial program 53.7%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 53.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 53.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

   8. Applied egg-rr 54.5%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

   10. Applied egg-rr 66.3%

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

   11. Taylor expanded in k around 0

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

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

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

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

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

       3. associate-*r/ N/A

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

   13. Simplified 99.7%

       \[\leadsto \frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{\color{blue}{k \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + 0.6666666666666666 \cdot \left(t \cdot t\right)\right)\right)}{\ell}\right)}} \]

   if 4 < k

   1. Initial program 63.1%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 63.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 29.2%

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

   7. Simplified 29.2%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

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

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

   9. Applied egg-rr 29.4%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   11. Applied egg-rr 79.9%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\frac{\frac{1}{\tan k}}{\sin k}}}{\ell}} \cdot \ell} \]
   12. Step-by-step derivation

       1. associate-/r/ N/A

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

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

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

   13. Applied egg-rr 83.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\left(t \cdot k\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 4:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{k \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.6666666666666666\right)\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{t}}{k}}{\sin k \cdot \left(k \cdot \tan k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t\_m}}{k \cdot \left(\frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.6666666666666666\right)\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e+133)
    (/
     (/ (/ 2.0 (/ (sin k) l)) t_m)
     (*
      k
      (+
       (/ (* 2.0 (* t_m t_m)) l)
       (/ (* k (* k (+ 1.0 (* (* t_m t_m) 0.6666666666666666)))) l))))
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e+133) {
		tmp = ((2.0 / (sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4d+133) then
        tmp = ((2.0d0 / (sin(k) / l)) / t_m) / (k * (((2.0d0 * (t_m * t_m)) / l) + ((k * (k * (1.0d0 + ((t_m * t_m) * 0.6666666666666666d0)))) / l)))
    else
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e+133) {
		tmp = ((2.0 / (Math.sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4e+133:
		tmp = ((2.0 / (math.sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)))
	else:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e+133)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) / l)) / t_m) / Float64(k * Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) / l) + Float64(Float64(k * Float64(k * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.6666666666666666)))) / l))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4e+133)
		tmp = ((2.0 / (sin(k) / l)) / t_m) / (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)));
	else
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e+133], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(k * N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + N[(N[(k * N[(k * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000001e133

   1. Initial program 61.5%

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

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

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   3. Simplified 57.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 68.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k}{\ell} \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

   8. Applied egg-rr 80.3%

      \[\leadsto \frac{2}{\frac{\sin k}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\ell}}}{\color{blue}{\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\ell}}}{t \cdot \color{blue}{\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{\color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}\right), \color{blue}{\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{\sin k}{\ell}}\right), t\right), \left(\color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right)} \cdot t\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell}\right)\right), t\right), \left(\left(\color{blue}{t} \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\sin k, \ell\right)\right), t\right), \left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \left(t \cdot \color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(t, \left(\frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell} \cdot \color{blue}{t}\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(t, \left(\left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \frac{\tan k}{\ell}\right) \cdot t\right)\right)\right) \]

   10. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\frac{\tan k}{\ell} \cdot t\right)\right)}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \color{blue}{\left(k \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2}}{\ell}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{2}}{\ell}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}}{\ell}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{2}\right), \ell\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}}{\ell}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({t}^{2}\right)\right), \ell\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot t\right)\right), \ell\right), \left(\frac{{k}^{\color{blue}{2}} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \left(\frac{{k}^{\color{blue}{2}} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\ell}\right)\right)\right)\right) \]

   13. Simplified 82.1%

       \[\leadsto \frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{\color{blue}{k \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + 0.6666666666666666 \cdot \left(t \cdot t\right)\right)\right)}{\ell}\right)}} \]

   if 4.0000000000000001e133 < t

   1. Initial program 44.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 39.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 39.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot k}\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       7. *-lowering-*.f64 83.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}} \cdot \frac{\ell}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{k \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.6666666666666666\right)\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(k \cdot \left(\frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.6666666666666666\right)\right)}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e+132)
    (/
     2.0
     (*
      (/ (sin k) l)
      (*
       t_m
       (*
        k
        (+
         (/ (* 2.0 (* t_m t_m)) l)
         (/ (* k (* k (+ 1.0 (* (* t_m t_m) 0.6666666666666666)))) l))))))
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e+132) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)))));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d+132) then
        tmp = 2.0d0 / ((sin(k) / l) * (t_m * (k * (((2.0d0 * (t_m * t_m)) / l) + ((k * (k * (1.0d0 + ((t_m * t_m) * 0.6666666666666666d0)))) / l)))))
    else
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e+132) {
		tmp = 2.0 / ((Math.sin(k) / l) * (t_m * (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)))));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8e+132:
		tmp = 2.0 / ((math.sin(k) / l) * (t_m * (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)))))
	else:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e+132)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(k * Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) / l) + Float64(Float64(k * Float64(k * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.6666666666666666)))) / l))))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e+132)
		tmp = 2.0 / ((sin(k) / l) * (t_m * (k * (((2.0 * (t_m * t_m)) / l) + ((k * (k * (1.0 + ((t_m * t_m) * 0.6666666666666666)))) / l)))));
	else
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e+132], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + N[(N[(k * N[(k * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.99999999999999993e132

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 68.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k}{\ell} \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

   8. Applied egg-rr 80.3%

      \[\leadsto \frac{2}{\frac{\sin k}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\color{blue}{\left(k \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}, t\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right), t\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2}}{\ell}\right), \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right), t\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{2}}{\ell}\right), \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right), t\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{2}\right), \ell\right), \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right), t\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({t}^{2}\right)\right), \ell\right), \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right), t\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot t\right)\right), \ell\right), \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right), t\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right), t\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \ell\right)\right)\right), t\right)\right)\right) \]

   11. Simplified 81.4%

       \[\leadsto \frac{2}{\frac{\sin k}{\ell} \cdot \left(\color{blue}{\left(k \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{\left(\left(1 + \left(t \cdot t\right) \cdot 0.6666666666666666\right) \cdot k\right) \cdot k}{\ell}\right)\right)} \cdot t\right)} \]

   if 7.99999999999999993e132 < t

   1. Initial program 44.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 39.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 39.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot k}\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       7. *-lowering-*.f64 83.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}} \cdot \frac{\ell}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t \cdot \left(k \cdot \left(\frac{2 \cdot \left(t \cdot t\right)}{\ell} + \frac{k \cdot \left(k \cdot \left(1 + \left(t \cdot t\right) \cdot 0.6666666666666666\right)\right)}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+78}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \ell}{k}}{t\_m}}{t\_m \cdot \left(\left(t\_m \cdot \frac{\tan k}{\ell}\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t\_m}}{t\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+78)
    (/
     (/ (/ (* 2.0 l) k) t_m)
     (* t_m (* (* t_m (/ (tan k) l)) (+ 2.0 (/ (/ (* k k) t_m) t_m)))))
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+78) {
		tmp = (((2.0 * l) / k) / t_m) / (t_m * ((t_m * (tan(k) / l)) * (2.0 + (((k * k) / t_m) / t_m))));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+78) then
        tmp = (((2.0d0 * l) / k) / t_m) / (t_m * ((t_m * (tan(k) / l)) * (2.0d0 + (((k * k) / t_m) / t_m))))
    else
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+78) {
		tmp = (((2.0 * l) / k) / t_m) / (t_m * ((t_m * (Math.tan(k) / l)) * (2.0 + (((k * k) / t_m) / t_m))));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e+78:
		tmp = (((2.0 * l) / k) / t_m) / (t_m * ((t_m * (math.tan(k) / l)) * (2.0 + (((k * k) / t_m) / t_m))))
	else:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+78)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / k) / t_m) / Float64(t_m * Float64(Float64(t_m * Float64(tan(k) / l)) * Float64(2.0 + Float64(Float64(Float64(k * k) / t_m) / t_m)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+78)
		tmp = (((2.0 * l) / k) / t_m) / (t_m * ((t_m * (tan(k) / l)) * (2.0 + (((k * k) / t_m) / t_m))));
	else
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+78], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000001e78

   1. Initial program 64.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 74.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k}{\ell} \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

   8. Applied egg-rr 87.2%

      \[\leadsto \frac{2}{\frac{\sin k}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\ell}}}{\color{blue}{\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\ell}}}{t \cdot \color{blue}{\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{\color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}\right), \color{blue}{\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{\sin k}{\ell}}\right), t\right), \left(\color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right)} \cdot t\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell}\right)\right), t\right), \left(\left(\color{blue}{t} \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\sin k, \ell\right)\right), t\right), \left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \left(t \cdot \color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(t, \left(\frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell} \cdot \color{blue}{t}\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right)\right), t\right), \mathsf{*.f64}\left(t, \left(\left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \frac{\tan k}{\ell}\right) \cdot t\right)\right)\right) \]

   10. Applied egg-rr 90.2%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{\sin k}{\ell}}}{t}}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\frac{\tan k}{\ell} \cdot t\right)\right)}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{k \cdot t}\right)}, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{\ell}{k}}{t}\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{\ell}{k}}{t}\right), \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{\ell}{k}\right), t\right), \mathsf{*.f64}\left(\color{blue}{t}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), t\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), t\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot 2\right), k\right), t\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

       7. *-lowering-*.f64 88.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 2\right), k\right), t\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \ell\right), t\right)\right)\right)\right) \]

   13. Simplified 88.9%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot 2}{k}}{t}}}{t \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\frac{\tan k}{\ell} \cdot t\right)\right)} \]

   if 1.00000000000000001e78 < (*.f64 l l)

   1. Initial program 51.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 49.7%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 59.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 68.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot k}\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       7. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 77.2%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}} \cdot \frac{\ell}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+78}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \ell}{k}}{t}}{t \cdot \left(\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(2 + \frac{\frac{k \cdot k}{t}}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-124}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{\left(t\_m \cdot k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-124)
    (/
     2.0
     (*
      (/ k l)
      (*
       t_m
       (* t_m (* t_m (/ (* (tan k) (+ 2.0 (/ (/ k t_m) (/ t_m k)))) l))))))
    (/ (/ l (* t_m k)) (* (* t_m k) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-124) {
		tmp = 2.0 / ((k / l) * (t_m * (t_m * (t_m * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)))));
	} else {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-124) then
        tmp = 2.0d0 / ((k / l) * (t_m * (t_m * (t_m * ((tan(k) * (2.0d0 + ((k / t_m) / (t_m / k)))) / l)))))
    else
        tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-124) {
		tmp = 2.0 / ((k / l) * (t_m * (t_m * (t_m * ((Math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)))));
	} else {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-124:
		tmp = 2.0 / ((k / l) * (t_m * (t_m * (t_m * ((math.tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)))))
	else:
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-124)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(t_m * Float64(t_m * Float64(t_m * Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))) / l))))));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) / Float64(Float64(t_m * k) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-124)
		tmp = 2.0 / ((k / l) * (t_m * (t_m * (t_m * ((tan(k) * (2.0 + ((k / t_m) / (t_m / k)))) / l)))));
	else
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-124], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999933e-125

   1. Initial program 65.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k}{\ell} \cdot \left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\left(\frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell} \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right) \]

   8. Applied egg-rr 92.7%

      \[\leadsto \frac{2}{\frac{\sin k}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\left(\frac{k}{\ell}\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), \mathsf{/.f64}\left(t, k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \ell\right)\right), t\right), t\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 91.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), \mathsf{/.f64}\left(t, k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \ell\right)\right), t\right)}, t\right)\right)\right) \]

   11. Simplified 91.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\ell}} \cdot \left(\left(\left(t \cdot \frac{\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \tan k}{\ell}\right) \cdot t\right) \cdot t\right)} \]

   if 9.99999999999999933e-125 < (*.f64 l l)

   1. Initial program 54.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 51.2%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 58.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 65.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 73.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 73.9%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\color{blue}{\ell}}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)} \]

       3. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\frac{\ell}{t \cdot k}}{\color{blue}{t \cdot k}} \]

       4. frac-times N/A

          \[\leadsto \frac{1 \cdot \frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot k\right)}} \]

       5. clear-num N/A

          \[\leadsto \frac{1 \cdot \frac{1}{\frac{t \cdot k}{\ell}}}{\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)} \]

       6. div-inv N/A

          \[\leadsto \frac{\frac{1}{\frac{t \cdot k}{\ell}}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       7. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot k\right)\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\color{blue}{t} \cdot k\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       15. *-lowering-*.f64 75.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   13. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-124}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left(t \cdot \left(t \cdot \frac{\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\left(t \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-125}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m}}{\frac{\frac{k \cdot k}{\frac{\frac{1}{\tan k}}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{\left(t\_m \cdot k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e-125)
    (* l (/ (/ 2.0 t_m) (/ (/ (* k k) (/ (/ 1.0 (tan k)) k)) l)))
    (/ (/ l (* t_m k)) (* (* t_m k) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-125) {
		tmp = l * ((2.0 / t_m) / (((k * k) / ((1.0 / tan(k)) / k)) / l));
	} else {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.5d-125) then
        tmp = l * ((2.0d0 / t_m) / (((k * k) / ((1.0d0 / tan(k)) / k)) / l))
    else
        tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-125) {
		tmp = l * ((2.0 / t_m) / (((k * k) / ((1.0 / Math.tan(k)) / k)) / l));
	} else {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.5e-125:
		tmp = l * ((2.0 / t_m) / (((k * k) / ((1.0 / math.tan(k)) / k)) / l))
	else:
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e-125)
		tmp = Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(k * k) / Float64(Float64(1.0 / tan(k)) / k)) / l)));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) / Float64(Float64(t_m * k) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.5e-125)
		tmp = l * ((2.0 / t_m) / (((k * k) / ((1.0 / tan(k)) / k)) / l));
	else
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-125], N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] / N[(N[(1.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.5e-125

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{{\sin k}^{2}}{\left({t}^{2} \cdot \cos k\right) \cdot \color{blue}{{\ell}^{2}}}\right)\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\frac{{\sin k}^{2}}{{t}^{2} \cdot \cos k}}{\color{blue}{{\ell}^{2}}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{\sin k}^{2}}{{t}^{2} \cdot \cos k}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{\sin k}^{2}}{\cos k \cdot {t}^{2}}\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{\frac{{\sin k}^{2}}{\cos k}}{{t}^{2}}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\sin k}^{2}}{\cos k}\right), \left({t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2}\right), \cos k\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \cos k\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \cos k\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 27.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right)\right) \]

   7. Simplified 27.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\left(t \cdot t\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   9. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell}}{\color{blue}{\ell}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell}} \cdot \color{blue}{\ell} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t}}{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell}}\right), \color{blue}{\ell}\right) \]

   11. Applied egg-rr 73.7%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\frac{\frac{1}{\tan k}}{\sin k}}}{\ell}} \cdot \ell} \]

   12. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{k}\right)\right)\right), \ell\right)\right), \ell\right) \]
   13. Step-by-step derivation

   14. Simplified 67.1%

       \[\leadsto \frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\frac{\frac{1}{\tan k}}{\color{blue}{k}}}}{\ell}} \cdot \ell \]

   if 7.5e-125 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 49.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 72.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\color{blue}{\ell}}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)} \]

       3. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\frac{\ell}{t \cdot k}}{\color{blue}{t \cdot k}} \]

       4. frac-times N/A

          \[\leadsto \frac{1 \cdot \frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot k\right)}} \]

       5. clear-num N/A

          \[\leadsto \frac{1 \cdot \frac{1}{\frac{t \cdot k}{\ell}}}{\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)} \]

       6. div-inv N/A

          \[\leadsto \frac{\frac{1}{\frac{t \cdot k}{\ell}}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       7. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot k\right)\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\color{blue}{t} \cdot k\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       15. *-lowering-*.f64 79.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   13. Applied egg-rr 79.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-125}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t}}{\frac{\frac{k \cdot k}{\frac{\frac{1}{\tan k}}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\left(t \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 18.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-128)
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k)))))
    (if (<= t_m 1.9e+145)
      (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
      (* (/ l t_m) (/ l (* (* t_m k) (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-128) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else if (t_m <= 1.9e+145) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.8d-128) then
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    else if (t_m <= 1.9d+145) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-128) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else if (t_m <= 1.9e+145) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.8e-128:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	elif t_m <= 1.9e+145:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.8e-128)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	elseif (t_m <= 1.9e+145)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.8e-128)
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	elseif (t_m <= 1.9e+145)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-128], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+145], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.79999999999999993e-128

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]

   if 7.79999999999999993e-128 < t < 1.90000000000000006e145

   1. Initial program 61.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 56.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 62.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 67.2%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

       3. times-frac N/A

          \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}} \]

       4. associate-/r* N/A

          \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot t\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(k \cdot t\right) \cdot t\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       13. *-lowering-*.f64 75.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   13. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 1.90000000000000006e145 < t

   1. Initial program 44.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 35.9%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 38.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 57.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 80.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 80.5%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m}}{\frac{\frac{k \cdot k}{\frac{1}{k \cdot k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{\left(t\_m \cdot k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-125)
    (* l (/ (/ 2.0 t_m) (/ (/ (* k k) (/ 1.0 (* k k))) l)))
    (/ (/ l (* t_m k)) (* (* t_m k) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-125) {
		tmp = l * ((2.0 / t_m) / (((k * k) / (1.0 / (k * k))) / l));
	} else {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.8d-125) then
        tmp = l * ((2.0d0 / t_m) / (((k * k) / (1.0d0 / (k * k))) / l))
    else
        tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-125) {
		tmp = l * ((2.0 / t_m) / (((k * k) / (1.0 / (k * k))) / l));
	} else {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.8e-125:
		tmp = l * ((2.0 / t_m) / (((k * k) / (1.0 / (k * k))) / l))
	else:
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.8e-125)
		tmp = Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(k * k) / Float64(1.0 / Float64(k * k))) / l)));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) / Float64(Float64(t_m * k) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.8e-125)
		tmp = l * ((2.0 / t_m) / (((k * k) / (1.0 / (k * k))) / l));
	else
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-125], N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] / N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999965e-125

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \left({t}^{2} \cdot \cos k\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{{\sin k}^{2}}{\left({t}^{2} \cdot \cos k\right) \cdot \color{blue}{{\ell}^{2}}}\right)\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\frac{{\sin k}^{2}}{{t}^{2} \cdot \cos k}}{\color{blue}{{\ell}^{2}}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{\sin k}^{2}}{{t}^{2} \cdot \cos k}\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{\sin k}^{2}}{\cos k \cdot {t}^{2}}\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{\frac{{\sin k}^{2}}{\cos k}}{{t}^{2}}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\sin k}^{2}}{\cos k}\right), \left({t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\sin k}^{2}\right), \cos k\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \cos k\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \cos k\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \left({t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \left(t \cdot t\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 27.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right)\right) \]

   7. Simplified 27.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\ell \cdot \ell}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\left(t \cdot t\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\cos k}}{t \cdot t}\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   9. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell \cdot \ell}}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell}}{\color{blue}{\ell}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell}} \cdot \color{blue}{\ell} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t}}{\frac{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{t \cdot t}{\sin k \cdot \tan k}}}{\ell}}\right), \color{blue}{\ell}\right) \]

   11. Applied egg-rr 73.7%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\frac{\frac{1}{\tan k}}{\sin k}}}{\ell}} \cdot \ell} \]

   12. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{1}{{k}^{2}}\right)}\right)\right), \ell\right)\right), \ell\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(1, \left({k}^{2}\right)\right)\right)\right), \ell\right)\right), \ell\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(1, \left(k \cdot k\right)\right)\right)\right), \ell\right)\right), \ell\right) \]

       3. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right)\right), \ell\right) \]

   14. Simplified 67.1%

       \[\leadsto \frac{\frac{2}{t}}{\frac{1 \cdot \frac{k \cdot k}{\color{blue}{\frac{1}{k \cdot k}}}}{\ell}} \cdot \ell \]

   if 7.79999999999999965e-125 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 49.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 72.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\color{blue}{\ell}}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)} \]

       3. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\frac{\ell}{t \cdot k}}{\color{blue}{t \cdot k}} \]

       4. frac-times N/A

          \[\leadsto \frac{1 \cdot \frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot k\right)}} \]

       5. clear-num N/A

          \[\leadsto \frac{1 \cdot \frac{1}{\frac{t \cdot k}{\ell}}}{\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)} \]

       6. div-inv N/A

          \[\leadsto \frac{\frac{1}{\frac{t \cdot k}{\ell}}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       7. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot k\right)\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\color{blue}{t} \cdot k\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       15. *-lowering-*.f64 79.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   13. Applied egg-rr 79.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t}}{\frac{\frac{k \cdot k}{\frac{1}{k \cdot k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\left(t \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-66}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{\left(t\_m \cdot k\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{\frac{\frac{\ell}{\frac{t\_m}{\ell}}}{t\_m \cdot \left(k \cdot k\right)}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-66)
    (/ (/ l (* t_m k)) (* (* t_m k) (/ t_m l)))
    (/ 1.0 (/ t_m (/ (/ l (/ t_m l)) (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-66) {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	} else {
		tmp = 1.0 / (t_m / ((l / (t_m / l)) / (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-66) then
        tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
    else
        tmp = 1.0d0 / (t_m / ((l / (t_m / l)) / (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-66) {
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	} else {
		tmp = 1.0 / (t_m / ((l / (t_m / l)) / (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e-66:
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l))
	else:
		tmp = 1.0 / (t_m / ((l / (t_m / l)) / (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e-66)
		tmp = Float64(Float64(l / Float64(t_m * k)) / Float64(Float64(t_m * k) * Float64(t_m / l)));
	else
		tmp = Float64(1.0 / Float64(t_m / Float64(Float64(l / Float64(t_m / l)) / Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e-66)
		tmp = (l / (t_m * k)) / ((t_m * k) * (t_m / l));
	else
		tmp = 1.0 / (t_m / ((l / (t_m / l)) / (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e-66], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$m / N[(N[(l / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999998e-67

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 66.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 66.6%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 74.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 74.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\color{blue}{\ell}}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)} \]

       3. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\frac{\ell}{t \cdot k}}{\color{blue}{t \cdot k}} \]

       4. frac-times N/A

          \[\leadsto \frac{1 \cdot \frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot k\right)}} \]

       5. clear-num N/A

          \[\leadsto \frac{1 \cdot \frac{1}{\frac{t \cdot k}{\ell}}}{\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)} \]

       6. div-inv N/A

          \[\leadsto \frac{\frac{1}{\frac{t \cdot k}{\ell}}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       7. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot k\right)\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(t \cdot k\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\color{blue}{t} \cdot k\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       15. *-lowering-*.f64 81.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   13. Applied egg-rr 81.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}} \]

   if 9.9999999999999998e-67 < k

   1. Initial program 60.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 66.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 73.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell}{\color{blue}{t}} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{t}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell\right)}\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)} \cdot \ell\right)\right)\right) \]

       6. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \left(\frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \left(\frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{t} \cdot \left(k \cdot k\right)}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t}\right), \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right), \left(t \cdot \left(k \cdot k\right)\right)\right)\right)\right) \]

       10. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\color{blue}{t} \cdot \left(k \cdot k\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{t} \cdot \left(k \cdot k\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(t \cdot \left(k \cdot k\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 73.4%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 73.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{\frac{\frac{\ell}{\frac{t}{\ell}}}{t \cdot \left(k \cdot k\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-66}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\left(t \cdot k\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{\frac{\frac{\ell}{\frac{t}{\ell}}}{t \cdot \left(k \cdot k\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.4% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.5e+91)
    (/ (* (/ l (* t_m k)) (/ l t_m)) (* t_m k))
    (/ (/ (* l l) (* t_m (* t_m (* k k)))) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e+91) {
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k);
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.5d+91) then
        tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k)
    else
        tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e+91) {
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k);
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.5e+91:
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k)
	else:
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.5e+91)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) * Float64(l / t_m)) / Float64(t_m * k));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.5e+91)
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k);
	else
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.5e+91], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.50000000000000001e91

   1. Initial program 58.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 68.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 75.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 75.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{t \cdot k} \cdot \frac{\color{blue}{\ell}}{t} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{\color{blue}{t \cdot k}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}\right), \color{blue}{\left(t \cdot k\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \left(\frac{\ell}{t}\right)\right), \left(\color{blue}{t} \cdot k\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\frac{\ell}{t}\right)\right), \left(t \cdot k\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(\frac{\ell}{t}\right)\right), \left(t \cdot k\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{\ell}{t}\right)\right), \left(t \cdot k\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(t \cdot k\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \color{blue}{t}\right)\right) \]

       10. *-lowering-*.f64 81.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right) \]

   13. Applied egg-rr 81.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}} \]

   if 3.50000000000000001e91 < k

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 58.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{t}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), t\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), t\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right) \]

      11. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right) \]

   9. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.6% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.2e+121)
    (/ (* (/ l (* t_m k)) (/ l t_m)) (* t_m k))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.2e+121) {
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k);
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.2d+121) then
        tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k)
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.2e+121) {
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k);
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.2e+121:
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k)
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.2e+121)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) * Float64(l / t_m)) / Float64(t_m * k));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.2e+121)
		tmp = ((l / (t_m * k)) * (l / t_m)) / (t_m * k);
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.2e+121], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.20000000000000016e121

   1. Initial program 58.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 67.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 74.4%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{t \cdot k} \cdot \frac{\color{blue}{\ell}}{t} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{\color{blue}{t \cdot k}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}\right), \color{blue}{\left(t \cdot k\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \left(\frac{\ell}{t}\right)\right), \left(\color{blue}{t} \cdot k\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\frac{\ell}{t}\right)\right), \left(t \cdot k\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(\frac{\ell}{t}\right)\right), \left(t \cdot k\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{\ell}{t}\right)\right), \left(t \cdot k\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(t \cdot k\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \color{blue}{t}\right)\right) \]

       10. *-lowering-*.f64 80.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right) \]

   13. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}} \]

   if 6.20000000000000016e121 < k

   1. Initial program 58.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 63.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 73.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.6% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-126}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m}}{k} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-126)
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k)))))
    (* (/ (/ (/ l (* t_m k)) t_m) k) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-126) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else {
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1d-126) then
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    else
        tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-126) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else {
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1e-126:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	else:
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-126)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(t_m * k)) / t_m) / k) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1e-126)
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	else
		tmp = (((l / (t_m * k)) / t_m) / k) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-126], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999995e-127

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]

   if 9.9999999999999995e-127 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 49.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 72.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t \cdot k}}{t \cdot k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t \cdot k}}{t}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot k}\right), t\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), t\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), t\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       7. *-lowering-*.f64 78.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), t\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   13. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k \cdot t}}{t}}{k}} \cdot \frac{\ell}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-126}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.6% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-128)
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k)))))
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-128) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.8d-128) then
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    else
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-128) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.8e-128:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	else:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.8e-128)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.8e-128)
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	else
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-128], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999993e-128

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]

   if 7.79999999999999993e-128 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 49.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 60.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot k}\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, t\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       7. *-lowering-*.f64 78.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}} \cdot \frac{\ell}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.7% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-156}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-156)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-156) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-156) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-156) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e-156:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e-156)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e-156)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e-156], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000004e-156

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 64.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

       5. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

   11. Applied egg-rr 73.3%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
   12. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

       3. times-frac N/A

          \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}} \]

       4. associate-/r* N/A

          \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot t\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(k \cdot t\right) \cdot t\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       13. *-lowering-*.f64 69.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   13. Applied egg-rr 69.6%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 1.00000000000000004e-156 < k

   1. Initial program 59.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

      14. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

   7. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

      11. /-lowering-/.f64 75.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 75.6%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-156}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.9% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (t_m * k)) * (l / (k * (t_m * t_m))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

3. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. unpow3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

   14. *-lowering-*.f64 58.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

7. Simplified 58.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   11. /-lowering-/.f64 68.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

9. Applied egg-rr 68.5%

   \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    2. unswap-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    5. *-lowering-*.f64 73.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

11. Applied egg-rr 73.5%

    \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
12. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

    2. associate-*r* N/A

       \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

    3. times-frac N/A

       \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}} \]

    4. associate-/r* N/A

       \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot k\right) \cdot t}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot t\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(k \cdot t\right) \cdot t\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{t}\right)\right)\right) \]

    13. *-lowering-*.f64 69.0%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

13. Applied egg-rr 69.0%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

14. Final simplification 69.0%

    \[\leadsto \frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)} \]
15. Add Preprocessing

Alternative 19: 62.6% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* k (* k (* t_m t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (k * (t_m * t_m))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (k * (k * (t_m * t_m))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (k * (t_m * t_m))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (k * (k * (t_m * t_m))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(k * Float64(t_m * t_m)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (k * (k * (t_m * t_m))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(k * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

3. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. unpow3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot {\color{blue}{k}}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left({t}^{2} \cdot t\right) \cdot {k}^{2}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{2} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{t}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right)\right) \]

   14. *-lowering-*.f64 58.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right)\right) \]

7. Simplified 58.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{\ell}{t}\right)\right) \]

   11. /-lowering-/.f64 68.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right) \]

9. Applied egg-rr 68.5%

   \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    2. unswap-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(t \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

    5. *-lowering-*.f64 73.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right) \]

11. Applied egg-rr 73.5%

    \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t} \]
12. Step-by-step derivation

    1. frac-times N/A

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t}} \]

    2. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t}\right)}\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}}\right)\right) \]

    5. swap-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

    9. swap-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\right)\right)\right) \]

    11. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(\left(t \cdot k\right) \cdot t\right) \cdot \color{blue}{k}\right)\right)\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot t\right)}\right)\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(t \cdot k\right) \cdot t\right)}\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(\left(k \cdot t\right) \cdot t\right)\right)\right)\right)\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 65.7%

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

13. Applied egg-rr 65.7%

    \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot \left(t \cdot t\right)\right)\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))