Toniolo and Linder, Equation (10+)

Average Error: 32.9 → 5.0

Time: 29.5s

Precision: binary64

Cost: 20937

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+14} \lor \neg \left(t \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{-\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2}}}{t \cdot \frac{k}{\ell}}\\ \end{array} \]

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 -1e+14) (not (<= t 8.2e-5)))
   (/
    2.0
    (/
     (* (* t (* (/ t l) (sin k))) (* (tan k) (- -2.0 (pow (/ k t) 2.0))))
     (/ (- l) t)))
   (* 2.0 (/ (* (/ l k) (/ (cos k) (pow (sin k) 2.0))) (* t (/ k l))))))

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 <= -1e+14) || !(t <= 8.2e-5)) {
		tmp = 2.0 / (((t * ((t / l) * sin(k))) * (tan(k) * (-2.0 - pow((k / t), 2.0)))) / (-l / t));
	} else {
		tmp = 2.0 * (((l / k) * (cos(k) / pow(sin(k), 2.0))) / (t * (k / l)));
	}
	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 <= (-1d+14)) .or. (.not. (t <= 8.2d-5))) then
        tmp = 2.0d0 / (((t * ((t / l) * sin(k))) * (tan(k) * ((-2.0d0) - ((k / t) ** 2.0d0)))) / (-l / t))
    else
        tmp = 2.0d0 * (((l / k) * (cos(k) / (sin(k) ** 2.0d0))) / (t * (k / l)))
    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 <= -1e+14) || !(t <= 8.2e-5)) {
		tmp = 2.0 / (((t * ((t / l) * Math.sin(k))) * (Math.tan(k) * (-2.0 - Math.pow((k / t), 2.0)))) / (-l / t));
	} else {
		tmp = 2.0 * (((l / k) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0))) / (t * (k / l)));
	}
	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 <= -1e+14) or not (t <= 8.2e-5):
		tmp = 2.0 / (((t * ((t / l) * math.sin(k))) * (math.tan(k) * (-2.0 - math.pow((k / t), 2.0)))) / (-l / t))
	else:
		tmp = 2.0 * (((l / k) * (math.cos(k) / math.pow(math.sin(k), 2.0))) / (t * (k / l)))
	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 <= -1e+14) || !(t <= 8.2e-5))
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(t / l) * sin(k))) * Float64(tan(k) * Float64(-2.0 - (Float64(k / t) ^ 2.0)))) / Float64(Float64(-l) / t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / (sin(k) ^ 2.0))) / Float64(t * Float64(k / l))));
	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 <= -1e+14) || ~((t <= 8.2e-5)))
		tmp = 2.0 / (((t * ((t / l) * sin(k))) * (tan(k) * (-2.0 - ((k / t) ^ 2.0)))) / (-l / t));
	else
		tmp = 2.0 * (((l / k) * (cos(k) / (sin(k) ^ 2.0))) / (t * (k / l)));
	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, -1e+14], N[Not[LessEqual[t, 8.2e-5]], $MachinePrecision]], N[(2.0 / N[(N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(-2.0 - N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-l) / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1e14 or 8.20000000000000009e-5 < t

   1. Initial program 22.8

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

   2. Simplified 22.8

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

      [Start] 22.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 22.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 22.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 15.7

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

   4. Applied egg-rr 6.1

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

   5. Applied egg-rr 3.6

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

   if -1e14 < t < 8.20000000000000009e-5

   1. Initial program 49.1

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

   2. Simplified 49.0

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

      [Start] 49.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 49.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 49.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 41.7

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

   4. Applied egg-rr 34.2

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

   5. Taylor expanded in t around 0 28.0

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

   6. Simplified 17.8

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

      [Start] 28.0	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 28.0	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 29.5	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 29.5	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 29.5	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 17.8	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   7. Applied egg-rr 7.2

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+14} \lor \neg \left(t \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{-\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2}}}{t \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.1
Cost	20872

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := t \cdot \frac{k}{\ell}\\ t_3 := \frac{-\ell}{t}\\ t_4 := \frac{\cos k}{{\sin k}^{2}}\\ \mathbf{if}\;k \leq -3.3 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_4}{t_2}\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + t_1\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_3}}\\ \mathbf{elif}\;k \leq 0.0021:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - t_1\right)\right)}{t_3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_4}{\frac{k}{\ell} \cdot t_2}\\ \end{array} \]

Alternative 2
Error	11.4
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := \frac{-\ell}{t}\\ \mathbf{if}\;k \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_2}}\\ \mathbf{elif}\;k \leq 0.00215:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.4
Cost	20620

\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := \frac{-\ell}{t}\\ \mathbf{if}\;k \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot t_1}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_2}}\\ \mathbf{elif}\;k \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \]

Alternative 4
Error	8.6
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ t_2 := \frac{-\ell}{t}\\ \mathbf{if}\;k \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_2}}\\ \mathbf{elif}\;k \leq 0.002:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	8.6
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2}}}{t \cdot \frac{k}{\ell}}\\ t_2 := \frac{-\ell}{t}\\ \mathbf{if}\;k \leq -1.02 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_2}}\\ \mathbf{elif}\;k \leq 0.00215:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	8.6
Cost	20620

\[\begin{array}{l} t_1 := t \cdot \frac{k}{\ell}\\ t_2 := \frac{\cos k}{{\sin k}^{2}}\\ t_3 := \frac{-\ell}{t}\\ \mathbf{if}\;k \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_3}}\\ \mathbf{elif}\;k \leq 0.00215:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{t_3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2}{\frac{k}{\ell} \cdot t_1}\\ \end{array} \]

Alternative 7
Error	11.5
Cost	14668

\[\begin{array}{l} t_1 := \frac{-\ell}{t}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{t_1}}\\ \mathbf{elif}\;k \leq 0.00215:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	12.0
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -1.02 \cdot 10^{-6} \lor \neg \left(k \leq 0.00075\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{\frac{-\ell}{t}}}\\ \end{array} \]

Alternative 9
Error	21.5
Cost	7817

\[\begin{array}{l} \mathbf{if}\;t \leq -440000000000 \lor \neg \left(t \leq 2.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(k \cdot -2\right)}{\frac{-\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 10
Error	21.5
Cost	7753

\[\begin{array}{l} \mathbf{if}\;t \leq -500000000000 \lor \neg \left(t \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{k + k}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 11
Error	24.6
Cost	7304

\[\begin{array}{l} t_1 := k \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{t_1}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{t_1}\right)\\ \end{array} \]

Alternative 12
Error	28.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-32} \lor \neg \left(t \leq 9 \cdot 10^{-83}\right):\\ \;\;\;\;\ell \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 13
Error	29.2
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-35} \lor \neg \left(t \leq 1.35 \cdot 10^{-29}\right):\\ \;\;\;\;\ell \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\\ \end{array} \]

Alternative 14
Error	29.1
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-35} \lor \neg \left(t \leq 3 \cdot 10^{-81}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 15
Error	29.0
Cost	1096

\[\begin{array}{l} t_1 := k \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{t_1}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t_1}}{t}\\ \end{array} \]

Alternative 16
Error	30.3
Cost	832

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

Alternative 17
Error	30.1
Cost	832

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

Reproduce

herbie shell --seed 2023027 
(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))))