Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 6.0

Time: 38.1s

Precision: binary64

Cost: 20752

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{{\sin k}^{2}}{\ell}\\ t_2 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k \cdot t}}{t_1}\\ \mathbf{if}\;k \leq -8.131690171837042 \cdot 10^{+243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \frac{\ell}{t}}{t_1}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{{\left(\frac{\frac{\ell}{{k}^{1.5}}}{t}\right)}^{2}}{\frac{1}{t}}\\ \mathbf{elif}\;k \leq 10^{-55}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{\frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ (pow (sin k) 2.0) l)))
        (t_2 (* 2.0 (/ (/ (* (cos k) l) (* k t)) t_1))))
   (if (<= k -8.131690171837042e+243)
     t_2
     (if (<= k -1e-160)
       (* 2.0 (/ (* (/ (cos k) k) (/ l t)) t_1))
       (if (<= k 5e-144)
         (* (/ 2.0 k) (/ (pow (/ (/ l (pow k 1.5)) t) 2.0) (/ 1.0 t)))
         (if (<= k 1e-55)
           (* 2.0 (/ (pow (/ l (* k (sin k))) 2.0) (/ t (cos k))))
           t_2))))))

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));
}

↓

double code(double t, double l, double k) {
	double t_1 = k * (pow(sin(k), 2.0) / l);
	double t_2 = 2.0 * (((cos(k) * l) / (k * t)) / t_1);
	double tmp;
	if (k <= -8.131690171837042e+243) {
		tmp = t_2;
	} else if (k <= -1e-160) {
		tmp = 2.0 * (((cos(k) / k) * (l / t)) / t_1);
	} else if (k <= 5e-144) {
		tmp = (2.0 / k) * (pow(((l / pow(k, 1.5)) / t), 2.0) / (1.0 / t));
	} else if (k <= 1e-55) {
		tmp = 2.0 * (pow((l / (k * sin(k))), 2.0) / (t / cos(k)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k * ((sin(k) ** 2.0d0) / l)
    t_2 = 2.0d0 * (((cos(k) * l) / (k * t)) / t_1)
    if (k <= (-8.131690171837042d+243)) then
        tmp = t_2
    else if (k <= (-1d-160)) then
        tmp = 2.0d0 * (((cos(k) / k) * (l / t)) / t_1)
    else if (k <= 5d-144) then
        tmp = (2.0d0 / k) * ((((l / (k ** 1.5d0)) / t) ** 2.0d0) / (1.0d0 / t))
    else if (k <= 1d-55) then
        tmp = 2.0d0 * (((l / (k * sin(k))) ** 2.0d0) / (t / cos(k)))
    else
        tmp = t_2
    end if
    code = tmp
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));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = k * (Math.pow(Math.sin(k), 2.0) / l);
	double t_2 = 2.0 * (((Math.cos(k) * l) / (k * t)) / t_1);
	double tmp;
	if (k <= -8.131690171837042e+243) {
		tmp = t_2;
	} else if (k <= -1e-160) {
		tmp = 2.0 * (((Math.cos(k) / k) * (l / t)) / t_1);
	} else if (k <= 5e-144) {
		tmp = (2.0 / k) * (Math.pow(((l / Math.pow(k, 1.5)) / t), 2.0) / (1.0 / t));
	} else if (k <= 1e-55) {
		tmp = 2.0 * (Math.pow((l / (k * Math.sin(k))), 2.0) / (t / Math.cos(k)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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))

↓

def code(t, l, k):
	t_1 = k * (math.pow(math.sin(k), 2.0) / l)
	t_2 = 2.0 * (((math.cos(k) * l) / (k * t)) / t_1)
	tmp = 0
	if k <= -8.131690171837042e+243:
		tmp = t_2
	elif k <= -1e-160:
		tmp = 2.0 * (((math.cos(k) / k) * (l / t)) / t_1)
	elif k <= 5e-144:
		tmp = (2.0 / k) * (math.pow(((l / math.pow(k, 1.5)) / t), 2.0) / (1.0 / t))
	elif k <= 1e-55:
		tmp = 2.0 * (math.pow((l / (k * math.sin(k))), 2.0) / (t / math.cos(k)))
	else:
		tmp = t_2
	return tmp

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

↓

function code(t, l, k)
	t_1 = Float64(k * Float64((sin(k) ^ 2.0) / l))
	t_2 = Float64(2.0 * Float64(Float64(Float64(cos(k) * l) / Float64(k * t)) / t_1))
	tmp = 0.0
	if (k <= -8.131690171837042e+243)
		tmp = t_2;
	elseif (k <= -1e-160)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / k) * Float64(l / t)) / t_1));
	elseif (k <= 5e-144)
		tmp = Float64(Float64(2.0 / k) * Float64((Float64(Float64(l / (k ^ 1.5)) / t) ^ 2.0) / Float64(1.0 / t)));
	elseif (k <= 1e-55)
		tmp = Float64(2.0 * Float64((Float64(l / Float64(k * sin(k))) ^ 2.0) / Float64(t / cos(k))));
	else
		tmp = t_2;
	end
	return tmp
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

↓

function tmp_2 = code(t, l, k)
	t_1 = k * ((sin(k) ^ 2.0) / l);
	t_2 = 2.0 * (((cos(k) * l) / (k * t)) / t_1);
	tmp = 0.0;
	if (k <= -8.131690171837042e+243)
		tmp = t_2;
	elseif (k <= -1e-160)
		tmp = 2.0 * (((cos(k) / k) * (l / t)) / t_1);
	elseif (k <= 5e-144)
		tmp = (2.0 / k) * ((((l / (k ^ 1.5)) / t) ^ 2.0) / (1.0 / t));
	elseif (k <= 1e-55)
		tmp = 2.0 * (((l / (k * sin(k))) ^ 2.0) / (t / cos(k)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
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]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -8.131690171837042e+243], t$95$2, If[LessEqual[k, -1e-160], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e-144], N[(N[(2.0 / k), $MachinePrecision] * N[(N[Power[N[(N[(l / N[Power[k, 1.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1e-55], N[(2.0 * N[(N[Power[N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -8.1316901718370416e243 or 9.99999999999999995e-56 < k

   1. Initial program 43.6

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

   2. Simplified 37.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 31 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 2 points increase in error, 2 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error

   3. Applied egg-rr 33.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\left(\tan k \cdot 0.5\right) \cdot \frac{\sin k}{{t}^{-3}}}}{\frac{k}{t}}}{\frac{k}{t}}} \]

   4. Taylor expanded in l around 0 18.9

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

   5. Simplified 15.5

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{t} \cdot \ell}{{\sin k}^{2}}\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (*.f64 (/.f64 l t) l) (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (*.f64 (/.f64 l t) l) (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 l t) (/.f64 l (pow.f64 (sin.f64 k) 2)))))): 8 points increase in error, 16 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 39 points increase in error, 8 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (pow.f64 l 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 20 points increase in error, 18 points decrease in error

   6. Applied egg-rr 4.7

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

   7. Taylor expanded in k around inf 4.4

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

   if -8.1316901718370416e243 < k < -9.9999999999999999e-161

   1. Initial program 49.7

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

   2. Simplified 41.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 31 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 2 points increase in error, 2 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error

   3. Applied egg-rr 35.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\left(\tan k \cdot 0.5\right) \cdot \frac{\sin k}{{t}^{-3}}}}{\frac{k}{t}}}{\frac{k}{t}}} \]

   4. Taylor expanded in l around 0 21.2

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

   5. Simplified 15.9

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{t} \cdot \ell}{{\sin k}^{2}}\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (*.f64 (/.f64 l t) l) (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (*.f64 (/.f64 l t) l) (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 l t) (/.f64 l (pow.f64 (sin.f64 k) 2)))))): 8 points increase in error, 16 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 39 points increase in error, 8 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (pow.f64 l 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 20 points increase in error, 18 points decrease in error

   6. Applied egg-rr 5.0

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

   if -9.9999999999999999e-161 < k < 4.9999999999999998e-144

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 31 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 2 points increase in error, 2 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error

   3. Applied egg-rr 63.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\left(\tan k \cdot 0.5\right) \cdot \frac{\sin k}{{t}^{-3}}}}{\frac{k}{t}}}{\frac{k}{t}}} \]

   4. Taylor expanded in k around 0 64.0

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

   5. Simplified 64.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{{k}^{3}}}{t \cdot t}}}{\frac{k}{t}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 (*.f64 l l) (pow.f64 k 3)) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (pow.f64 k 3)) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (pow.f64 l 2) (pow.f64 k 3)) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 k 3) (pow.f64 t 2))))): 6 points increase in error, 6 points decrease in error

   6. Applied egg-rr 38.8

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

   if 4.9999999999999998e-144 < k < 9.99999999999999995e-56

   1. Initial program 63.6

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

   2. Simplified 57.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 31 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 2 points increase in error, 2 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in k around inf 38.8

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

   4. Applied egg-rr 51.0

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

   5. Applied egg-rr 11.0

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

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.131690171837042 \cdot 10^{+243}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \ell}{k \cdot t}}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \frac{\ell}{t}}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{{\left(\frac{\frac{\ell}{{k}^{1.5}}}{t}\right)}^{2}}{\frac{1}{t}}\\ \mathbf{elif}\;k \leq 10^{-55}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{\frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \ell}{k \cdot t}}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.0
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{k \cdot \frac{t}{\ell}}}{k \cdot {\sin k}^{2}}\right)\\ t_2 := k \cdot \sin k\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+70}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \frac{\ell}{t}}{k \cdot \frac{0.5 + -0.5 \cdot \cos \left(k + k\right)}{\ell}}\\ \mathbf{elif}\;\ell \leq -3.5616211022466916 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.4242402514969963 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{t_2}\right)}^{2}}{\frac{t}{\cos k}}\\ \mathbf{elif}\;\ell \leq 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{{\left(\frac{t_2}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 2
Error	5.8
Cost	20752

\[\begin{array}{l} t_1 := \frac{\ell}{{\sin k}^{2}}\\ t_2 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{\frac{k}{\frac{\ell}{t}}} \cdot t_1\right)\\ t_3 := k \cdot \sin k\\ \mathbf{if}\;k \leq -5.1173293913825904 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\frac{t_3}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-110}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\frac{\ell \cdot -0.5}{t} + \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\right)\\ \mathbf{elif}\;k \leq 1.5113023476696617 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{t_3}\right)}^{2} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]

Alternative 3
Error	6.0
Cost	20752

\[\begin{array}{l} t_1 := k \cdot \frac{{\sin k}^{2}}{\ell}\\ t_2 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k \cdot t}}{t_1}\\ \mathbf{if}\;k \leq -5.545037427020772 \cdot 10^{+236}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t}}{\frac{k}{\cos k}}}{t_1}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{{\left(\frac{\frac{\ell}{{k}^{1.5}}}{t}\right)}^{2}}{\frac{1}{t}}\\ \mathbf{elif}\;k \leq 10^{-55}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{\frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	6.6
Cost	20488

\[\begin{array}{l} t_1 := k \cdot \sin k\\ \mathbf{if}\;k \leq -5.1173293913825904 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\frac{t_1}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.271589758824847 \cdot 10^{+136}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{t_1}\right)}^{2} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]

Alternative 5
Error	6.5
Cost	20488

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k \cdot t}}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{if}\;\ell \leq -2.1034357476450325 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.191260484378666 \cdot 10^{-247}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{\frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.6
Cost	14412

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\frac{\ell}{0.5}}{\tan k}}{k} \cdot \frac{\ell}{\sin k \cdot \left(t \cdot t\right)}}{\frac{k}{t}}\\ t_2 := k \cdot \frac{{\sin k}^{2}}{\ell}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-129}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t}}{t_2}\\ \mathbf{elif}\;t \leq 6.720006172195796 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot t}}{t_2}\\ \end{array} \]

Alternative 7
Error	6.6
Cost	14408

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\cos k}{k} \cdot \frac{\ell}{t}}{k \cdot \frac{0.5 + -0.5 \cdot \cos \left(k + k\right)}{\ell}}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t}}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	23.3
Cost	13696

\[2 \cdot \frac{\frac{\frac{\ell}{k}}{t}}{k \cdot \frac{{\sin k}^{2}}{\ell}} \]

Alternative 9
Error	25.8
Cost	7488

\[2 \cdot \frac{\frac{\cos k}{k} \cdot \frac{\ell}{t}}{k \cdot \frac{k \cdot k}{\ell}} \]

Alternative 10
Error	25.5
Cost	7488

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

Alternative 11
Error	26.7
Cost	960

\[\frac{\frac{\frac{2}{k}}{k} \cdot \frac{\ell}{k \cdot \frac{t}{\ell}}}{k} \]

Alternative 12
Error	26.0
Cost	960

\[\frac{\frac{2}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}}{k} \]

Alternative 13
Error	26.1
Cost	960

\[\frac{\frac{2}{k}}{k \cdot \frac{\frac{t}{\ell} \cdot \left(k \cdot k\right)}{\ell}} \]

Alternative 14
Error	25.9
Cost	960

\[\frac{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{2}{t}}{k}}{k \cdot k} \]

Reproduce

herbie shell --seed 2022290 
(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))))