Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 5.9

Time: 26.9s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-19} \lor \neg \left(t \leq 1.35 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\cos k}{{\sin k}^{2}}\right)}{t \cdot k}\\ \end{array} \]

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 -7.8e-19) (not (<= t 1.35e-111)))
   (/
    2.0
    (/
     (* (/ t l) (* t (sin k)))
     (/ (/ l t) (* (tan k) (+ 2.0 (/ k (* t (/ t k))))))))
   (* 2.0 (/ (* (/ l k) (* l (/ (cos k) (pow (sin k) 2.0)))) (* 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 <= -7.8e-19) || !(t <= 1.35e-111)) {
		tmp = 2.0 / (((t / l) * (t * sin(k))) / ((l / t) / (tan(k) * (2.0 + (k / (t * (t / k)))))));
	} else {
		tmp = 2.0 * (((l / k) * (l * (cos(k) / pow(sin(k), 2.0)))) / (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 <= (-7.8d-19)) .or. (.not. (t <= 1.35d-111))) then
        tmp = 2.0d0 / (((t / l) * (t * sin(k))) / ((l / t) / (tan(k) * (2.0d0 + (k / (t * (t / k)))))))
    else
        tmp = 2.0d0 * (((l / k) * (l * (cos(k) / (sin(k) ** 2.0d0)))) / (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 <= -7.8e-19) || !(t <= 1.35e-111)) {
		tmp = 2.0 / (((t / l) * (t * Math.sin(k))) / ((l / t) / (Math.tan(k) * (2.0 + (k / (t * (t / k)))))));
	} else {
		tmp = 2.0 * (((l / k) * (l * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)))) / (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 <= -7.8e-19) or not (t <= 1.35e-111):
		tmp = 2.0 / (((t / l) * (t * math.sin(k))) / ((l / t) / (math.tan(k) * (2.0 + (k / (t * (t / k)))))))
	else:
		tmp = 2.0 * (((l / k) * (l * (math.cos(k) / math.pow(math.sin(k), 2.0)))) / (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 <= -7.8e-19) || !(t <= 1.35e-111))
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(t * sin(k))) / Float64(Float64(l / t) / Float64(tan(k) * Float64(2.0 + Float64(k / Float64(t * Float64(t / k))))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l * Float64(cos(k) / (sin(k) ^ 2.0)))) / 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 <= -7.8e-19) || ~((t <= 1.35e-111)))
		tmp = 2.0 / (((t / l) * (t * sin(k))) / ((l / t) / (tan(k) * (2.0 + (k / (t * (t / k)))))));
	else
		tmp = 2.0 * (((l / k) * (l * (cos(k) / (sin(k) ^ 2.0)))) / (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, -7.8e-19], N[Not[LessEqual[t, 1.35e-111]], $MachinePrecision]], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l / t), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.7999999999999999e-19 or 1.34999999999999994e-111 < t

   1. Initial program 23.5

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

   2. Simplified 23.5

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

      [Start] 23.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 23.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 23.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 16.5

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

   4. Applied egg-rr 7.7

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

   5. Applied egg-rr 3.8

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

   6. Applied egg-rr 3.8

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

   if -7.7999999999999999e-19 < t < 1.34999999999999994e-111

   1. Initial program 56.9

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

   2. Simplified 56.9

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

      [Start] 56.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 56.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 56.8	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 56.8	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 57.1	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 57.4	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 56.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 56.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 56.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 56.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Taylor expanded in t around 0 28.2

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

   4. Simplified 30.1

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

      [Start] 28.2	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 30.1	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 30.1	\[ 2 \cdot \frac{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 30.1	\[ 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

   5. Applied egg-rr 25.9

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

   6. Applied egg-rr 11.1

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-19} \lor \neg \left(t \leq 1.35 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\cos k}{{\sin k}^{2}}\right)}{t \cdot k}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	20620

\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := 2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_1}\right)\\ \mathbf{if}\;k \leq -1.12 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 3500000000:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell \cdot \cos k}{t_1 \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	7.8
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-85} \lor \neg \left(t \leq 1.35 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3
Error	6.3
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-96} \lor \neg \left(t \leq 1.35 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4
Error	11.0
Cost	14665

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-104} \lor \neg \left(t \leq 3.1 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \end{array} \]

Alternative 5
Error	11.0
Cost	14665

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-104} \lor \neg \left(t \leq 1.9 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \end{array} \]

Alternative 6
Error	16.0
Cost	14473

\[\begin{array}{l} \mathbf{if}\;k \leq -150000000 \lor \neg \left(k \leq 15000000\right):\\ \;\;\;\;2 \cdot \left(\frac{2 \cdot \left(\frac{\ell}{t} \cdot \cos k\right)}{1 - \cos \left(k + k\right)} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]

Alternative 7
Error	15.9
Cost	14473

\[\begin{array}{l} \mathbf{if}\;k \leq -430000000 \lor \neg \left(k \leq 30000000\right):\\ \;\;\;\;2 \cdot \left(\frac{2 \cdot \left(\frac{\ell}{t} \cdot \cos k\right)}{1 - \cos \left(k + k\right)} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\ \end{array} \]

Alternative 8
Error	16.5
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -18000000 \lor \neg \left(k \leq 1250000000\right):\\ \;\;\;\;2 \cdot \left(\frac{2 \cdot \left(\frac{\ell}{t} \cdot \cos k\right)}{1 - \cos \left(k + k\right)} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 0.5}{t \cdot k}}}\\ \end{array} \]

Alternative 9
Error	19.4
Cost	7753

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-96} \lor \neg \left(t \leq 3.6 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 0.5}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 10
Error	24.4
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -6.3 \cdot 10^{-71} \lor \neg \left(t \leq 4.2 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 11
Error	24.1
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-96} \lor \neg \left(t \leq 8.2 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 12
Error	24.5
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-71}:\\ \;\;\;\;\ell \cdot \frac{t_1}{k}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \end{array} \]

Alternative 13
Error	24.5
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 14
Error	31.7
Cost	1225

\[\begin{array}{l} \mathbf{if}\;k \leq -900000000 \lor \neg \left(k \leq 2.5 \cdot 10^{-21}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\frac{k \cdot k}{\ell}}}{t \cdot t}}{t}\\ \end{array} \]

Alternative 15
Error	32.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\frac{k \cdot k}{\ell}}}{t \cdot t}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 16
Error	34.7
Cost	832

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

Alternative 17
Error	34.7
Cost	832

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

Reproduce

herbie shell --seed 2023012 
(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))))