Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 4.5

Time: 37.9s

Precision: binary64

Cost: 21001

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-83} \lor \neg \left(t \leq 8.8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{1}{\left(t \cdot \frac{t \cdot \tan k}{\ell}\right) \cdot \left(\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{2}{\sin k}} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{{\sin k}^{2} \cdot \frac{k}{\ell}}}{t \cdot k}\right)\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -8e-83) (not (<= t 8.8e-49)))
   (/
    1.0
    (*
     (* t (/ (* t (tan k)) l))
     (* (/ (+ 2.0 (pow (/ k t) 2.0)) (/ 2.0 (sin k))) (/ t l))))
   (* 2.0 (* l (/ (/ (cos k) (* (pow (sin k) 2.0) (/ k l))) (* t k))))))

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 tmp;
	if ((t <= -8e-83) || !(t <= 8.8e-49)) {
		tmp = 1.0 / ((t * ((t * tan(k)) / l)) * (((2.0 + pow((k / t), 2.0)) / (2.0 / sin(k))) * (t / l)));
	} else {
		tmp = 2.0 * (l * ((cos(k) / (pow(sin(k), 2.0) * (k / l))) / (t * k)));
	}
	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) :: tmp
    if ((t <= (-8d-83)) .or. (.not. (t <= 8.8d-49))) then
        tmp = 1.0d0 / ((t * ((t * tan(k)) / l)) * (((2.0d0 + ((k / t) ** 2.0d0)) / (2.0d0 / sin(k))) * (t / l)))
    else
        tmp = 2.0d0 * (l * ((cos(k) / ((sin(k) ** 2.0d0) * (k / l))) / (t * k)))
    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 tmp;
	if ((t <= -8e-83) || !(t <= 8.8e-49)) {
		tmp = 1.0 / ((t * ((t * Math.tan(k)) / l)) * (((2.0 + Math.pow((k / t), 2.0)) / (2.0 / Math.sin(k))) * (t / l)));
	} else {
		tmp = 2.0 * (l * ((Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * (k / l))) / (t * k)));
	}
	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):
	tmp = 0
	if (t <= -8e-83) or not (t <= 8.8e-49):
		tmp = 1.0 / ((t * ((t * math.tan(k)) / l)) * (((2.0 + math.pow((k / t), 2.0)) / (2.0 / math.sin(k))) * (t / l)))
	else:
		tmp = 2.0 * (l * ((math.cos(k) / (math.pow(math.sin(k), 2.0) * (k / l))) / (t * k)))
	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)
	tmp = 0.0
	if ((t <= -8e-83) || !(t <= 8.8e-49))
		tmp = Float64(1.0 / Float64(Float64(t * Float64(Float64(t * tan(k)) / l)) * Float64(Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) / Float64(2.0 / sin(k))) * Float64(t / l))));
	else
		tmp = Float64(2.0 * Float64(l * Float64(Float64(cos(k) / Float64((sin(k) ^ 2.0) * Float64(k / l))) / Float64(t * k))));
	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)
	tmp = 0.0;
	if ((t <= -8e-83) || ~((t <= 8.8e-49)))
		tmp = 1.0 / ((t * ((t * tan(k)) / l)) * (((2.0 + ((k / t) ^ 2.0)) / (2.0 / sin(k))) * (t / l)));
	else
		tmp = 2.0 * (l * ((cos(k) / ((sin(k) ^ 2.0) * (k / l))) / (t * k)));
	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_] := If[Or[LessEqual[t, -8e-83], N[Not[LessEqual[t, 8.8e-49]], $MachinePrecision]], N[(1.0 / N[(N[(t * N[(N[(t * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.0000000000000003e-83 or 8.79999999999999959e-49 < t

   1. Initial program 22.7

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

   2. Simplified 27.4

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

      [Start] 22.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.7	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 22.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 22.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 22.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 27.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 27.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 27.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 27.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      distribute-lft-in [=>] 36.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right) + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      associate-*l* [<=] 36.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-+r+ [=>] 36.3	\[ \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 36.9	\[ \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} + \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      associate-*l* [=>] 37.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right) + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-out [=>] 37.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k + \sin k \cdot \tan k\right)} + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-out [=>] 37.0	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \left(\tan k + \tan k\right)\right)} + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      count-2 [=>] 37.0	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot \tan k\right)}\right) + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [<=] 27.4	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(2 \cdot \tan k\right) + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      distribute-lft-in [<=] 27.4	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \left(2 \cdot \tan k + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      *-commutative [=>] 27.4	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(2 \cdot \tan k + \color{blue}{{\left(\frac{k}{t}\right)}^{2} \cdot \tan k}\right)\right)} \]
      distribute-rgt-in [<=] 27.4	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 21.7

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

   4. Applied egg-rr 18.1

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

   5. Applied egg-rr 11.1

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

   6. Applied egg-rr 4.6

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

   7. Simplified 3.7

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

      [Start] 4.6	\[ {\left(\left(\frac{t \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{2}\right) \cdot \frac{t}{\ell}\right)}^{-1} \]
      unpow-1 [=>] 4.6	\[ \color{blue}{\frac{1}{\left(\frac{t \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{2}\right) \cdot \frac{t}{\ell}}} \]
      associate-*l* [=>] 3.7	\[ \frac{1}{\color{blue}{\frac{t \cdot \tan k}{\frac{\ell}{t}} \cdot \left(\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{2} \cdot \frac{t}{\ell}\right)}} \]
      associate-/r/ [=>] 3.7	\[ \frac{1}{\color{blue}{\left(\frac{t \cdot \tan k}{\ell} \cdot t\right)} \cdot \left(\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{2} \cdot \frac{t}{\ell}\right)} \]
      associate-/l* [=>] 3.7	\[ \frac{1}{\left(\frac{t \cdot \tan k}{\ell} \cdot t\right) \cdot \left(\color{blue}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{2}{\sin k}}} \cdot \frac{t}{\ell}\right)} \]

   if -8.0000000000000003e-83 < t < 8.79999999999999959e-49

   1. Initial program 56.7

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

   2. Simplified 56.7

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

      [Start] 56.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 56.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 56.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 25.4

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

   4. Simplified 15.8

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

      [Start] 25.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 25.5	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 25.5	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 27.1	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 27.1	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 27.1	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 15.8	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

   5. Applied egg-rr 11.4

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

   6. Simplified 6.5

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

      [Start] 11.4	\[ 2 \cdot \left(\frac{1}{{\sin k}^{2} \cdot \frac{k}{\ell}} \cdot \frac{\cos k}{\frac{t}{\ell} \cdot k}\right) \]
      associate-*r/ [=>] 11.4	\[ 2 \cdot \color{blue}{\frac{\frac{1}{{\sin k}^{2} \cdot \frac{k}{\ell}} \cdot \cos k}{\frac{t}{\ell} \cdot k}} \]
      associate-*l/ [=>] 4.5	\[ 2 \cdot \frac{\frac{1}{{\sin k}^{2} \cdot \frac{k}{\ell}} \cdot \cos k}{\color{blue}{\frac{t \cdot k}{\ell}}} \]
      associate-/r/ [=>] 6.5	\[ 2 \cdot \color{blue}{\left(\frac{\frac{1}{{\sin k}^{2} \cdot \frac{k}{\ell}} \cdot \cos k}{t \cdot k} \cdot \ell\right)} \]
      associate-*l/ [=>] 6.5	\[ 2 \cdot \left(\frac{\color{blue}{\frac{1 \cdot \cos k}{{\sin k}^{2} \cdot \frac{k}{\ell}}}}{t \cdot k} \cdot \ell\right) \]
      *-lft-identity [=>] 6.5	\[ 2 \cdot \left(\frac{\frac{\color{blue}{\cos k}}{{\sin k}^{2} \cdot \frac{k}{\ell}}}{t \cdot k} \cdot \ell\right) \]
      *-commutative [=>] 6.5	\[ 2 \cdot \left(\frac{\frac{\cos k}{{\sin k}^{2} \cdot \frac{k}{\ell}}}{\color{blue}{k \cdot t}} \cdot \ell\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-83} \lor \neg \left(t \leq 8.8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{1}{\left(t \cdot \frac{t \cdot \tan k}{\ell}\right) \cdot \left(\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{2}{\sin k}} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{{\sin k}^{2} \cdot \frac{k}{\ell}}}{t \cdot k}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.1
Cost	21136

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ t_2 := 2 \cdot \left(\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{k}\right)\\ t_3 := \frac{\ell}{t} \cdot \frac{\frac{2}{t \cdot \frac{t}{\ell}}}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}\\ \mathbf{if}\;k \leq -8 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-136}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+114}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	8.8
Cost	21136

\[\begin{array}{l} t_1 := \tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\\ t_2 := 2 \cdot \left(\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{k}\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -3.45 \cdot 10^{+127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{t_1}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\ell}{t_3 \cdot \frac{k}{\frac{\ell}{{t_3}^{2}}}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+120}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{t \cdot \frac{t}{\ell}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	8.4
Cost	21136

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{k}\right)\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\\ \mathbf{if}\;k \leq -1.8 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4.05 \cdot 10^{-137}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \frac{\ell}{t}}{\left(t \cdot \tan k\right) \cdot t_3}}{t}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-151}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{t \cdot \frac{t}{\ell}}}{\tan k \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.6
Cost	21136

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ t_2 := 2 \cdot \left(\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{k}\right)\\ t_3 := \frac{\frac{\ell}{\frac{t \cdot \tan k}{\frac{\ell}{t}} \cdot \frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}{2}}}{t}\\ \mathbf{if}\;k \leq -3.75 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.6 \cdot 10^{-181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-208}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{elif}\;k \leq 3.05 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	14.7
Cost	20620

\[\begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 6
Error	12.0
Cost	20620

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 7
Error	11.9
Cost	20620

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+156}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 8
Error	11.9
Cost	20620

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{{\sin k}^{2} \cdot \frac{k}{\ell}}}{t \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 9
Error	13.5
Cost	20492

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ t_2 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -7 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.72 \cdot 10^{-102}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{2}{\tan k \cdot \left(2 \cdot \sin k\right)}}{t}\\ \mathbf{elif}\;k \leq 0.00092:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	13.8
Cost	14540

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-102}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{2}{\tan k \cdot \left(2 \cdot \sin k\right)}}{t}\\ \mathbf{elif}\;k \leq 0.0015:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	19.8
Cost	13964

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 12
Error	20.1
Cost	13644

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 13
Error	19.5
Cost	8336

\[\begin{array}{l} t_1 := \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ t_2 := \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-171}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	21.9
Cost	7760

\[\begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{-\ell}{t_1}}{-t}\\ \mathbf{elif}\;t \leq -8.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-108}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{-1}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(-{t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{t_1}\\ \end{array} \]

Alternative 15
Error	21.9
Cost	7568

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{-\ell}{t_2}}{-t}\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{t_1}{t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_1}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{t_2}\\ \end{array} \]

Alternative 16
Error	21.9
Cost	7436

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ t_2 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{-\ell}{t_2}}{-t}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{t_2}\\ \end{array} \]

Alternative 17
Error	25.0
Cost	1490

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+120} \lor \neg \left(t \leq -8.8 \cdot 10^{-107} \lor \neg \left(t \leq 7.4 \cdot 10^{-171}\right) \land t \leq 2.1 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{-\ell}{k \cdot \left(t \cdot k\right)}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ \end{array} \]

Alternative 18
Error	25.0
Cost	1488

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ t_2 := k \cdot \left(t \cdot k\right)\\ t_3 := \frac{\frac{\ell}{t} \cdot \frac{-\ell}{t_2}}{-t}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-171}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t_2}\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 19
Error	24.6
Cost	1488

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ t_2 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{-\ell}{t_2}}{-t}\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-171}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t_2}\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{t_2}\\ \end{array} \]

Alternative 20
Error	24.2
Cost	1488

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ t_2 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{-\ell}{t_2}}{-t}\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \frac{\ell}{t}}{2 \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\\ \mathbf{elif}\;t \leq 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{t_2}\\ \end{array} \]

Alternative 21
Error	29.4
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-34} \lor \neg \left(t \leq 8.6 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t \cdot t}\\ \end{array} \]

Alternative 22
Error	28.1
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\ell \leq -45 \lor \neg \left(\ell \leq 2.6 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ \end{array} \]

Alternative 23
Error	32.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t \cdot t}\\ \end{array} \]

Alternative 24
Error	29.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;k \leq -4.6 \cdot 10^{-157}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k}}{t \cdot k}}{t \cdot t}\\ \end{array} \]

Alternative 25
Error	31.4
Cost	832

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

Reproduce

herbie shell --seed 2022364 
(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))))