Toniolo and Linder, Equation (10+)

Average Error: 32.1 → 5.8

Time: 35.9s

Precision: binary64

Cost: 21001

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-41} \lor \neg \left(t \leq 1.05 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{t}{\ell}} \cdot \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \frac{-t}{\ell \cdot \left(-\cos k\right)}\right)}\\ \end{array} \]

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 -1.35e-41) (not (<= t 1.05e-24)))
   (*
    (/ (/ (/ 1.0 (sin k)) (+ 2.0 (pow (/ k t) 2.0))) (/ t l))
    (/ (/ 2.0 (* t (tan k))) (/ t l)))
   (/
    2.0
    (* (* (/ k l) (pow (sin k) 2.0)) (* k (/ (- t) (* l (- (cos 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 <= -1.35e-41) || !(t <= 1.05e-24)) {
		tmp = (((1.0 / sin(k)) / (2.0 + pow((k / t), 2.0))) / (t / l)) * ((2.0 / (t * tan(k))) / (t / l));
	} else {
		tmp = 2.0 / (((k / l) * pow(sin(k), 2.0)) * (k * (-t / (l * -cos(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 <= (-1.35d-41)) .or. (.not. (t <= 1.05d-24))) then
        tmp = (((1.0d0 / sin(k)) / (2.0d0 + ((k / t) ** 2.0d0))) / (t / l)) * ((2.0d0 / (t * tan(k))) / (t / l))
    else
        tmp = 2.0d0 / (((k / l) * (sin(k) ** 2.0d0)) * (k * (-t / (l * -cos(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 <= -1.35e-41) || !(t <= 1.05e-24)) {
		tmp = (((1.0 / Math.sin(k)) / (2.0 + Math.pow((k / t), 2.0))) / (t / l)) * ((2.0 / (t * Math.tan(k))) / (t / l));
	} else {
		tmp = 2.0 / (((k / l) * Math.pow(Math.sin(k), 2.0)) * (k * (-t / (l * -Math.cos(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 <= -1.35e-41) or not (t <= 1.05e-24):
		tmp = (((1.0 / math.sin(k)) / (2.0 + math.pow((k / t), 2.0))) / (t / l)) * ((2.0 / (t * math.tan(k))) / (t / l))
	else:
		tmp = 2.0 / (((k / l) * math.pow(math.sin(k), 2.0)) * (k * (-t / (l * -math.cos(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 <= -1.35e-41) || !(t <= 1.05e-24))
		tmp = Float64(Float64(Float64(Float64(1.0 / sin(k)) / Float64(2.0 + (Float64(k / t) ^ 2.0))) / Float64(t / l)) * Float64(Float64(2.0 / Float64(t * tan(k))) / Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * (sin(k) ^ 2.0)) * Float64(k * Float64(Float64(-t) / Float64(l * Float64(-cos(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 <= -1.35e-41) || ~((t <= 1.05e-24)))
		tmp = (((1.0 / sin(k)) / (2.0 + ((k / t) ^ 2.0))) / (t / l)) * ((2.0 / (t * tan(k))) / (t / l));
	else
		tmp = 2.0 / (((k / l) * (sin(k) ^ 2.0)) * (k * (-t / (l * -cos(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, -1.35e-41], N[Not[LessEqual[t, 1.05e-24]], $MachinePrecision]], N[(N[(N[(N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(t * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k * N[((-t) / N[(l * (-N[Cos[k], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e-41 or 1.05e-24 < t

   1. Initial program 22.3

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

   2. Simplified 27.6

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

      [Start] 22.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 22.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 22.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 22.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 27.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 27.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 27.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 27.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 21.2

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

   4. Applied egg-rr 18.2

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

   5. Simplified 9.2

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

      [Start] 18.2	\[ {\left(\frac{{\left(\frac{t}{\ell}\right)}^{2} \cdot t}{2} \cdot \left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)\right)}^{-1} \]
      unpow-1 [=>] 18.2	\[ \color{blue}{\frac{1}{\frac{{\left(\frac{t}{\ell}\right)}^{2} \cdot t}{2} \cdot \left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)}} \]
      associate-*r* [=>] 11.8	\[ \frac{1}{\color{blue}{\left(\frac{{\left(\frac{t}{\ell}\right)}^{2} \cdot t}{2} \cdot \tan k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}} \]
      *-commutative [=>] 11.8	\[ \frac{1}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \left(\frac{{\left(\frac{t}{\ell}\right)}^{2} \cdot t}{2} \cdot \tan k\right)}} \]
      *-commutative [=>] 11.8	\[ \frac{1}{\color{blue}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\frac{{\left(\frac{t}{\ell}\right)}^{2} \cdot t}{2} \cdot \tan k\right)} \]
      associate-*l/ [=>] 11.8	\[ \frac{1}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{\left({\left(\frac{t}{\ell}\right)}^{2} \cdot t\right) \cdot \tan k}{2}}} \]
      associate-*l* [=>] 9.2	\[ \frac{1}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot \tan k\right)}}{2}} \]

   6. Applied egg-rr 2.7

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

   if -1.35e-41 < t < 1.05e-24

   1. Initial program 52.8

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

   2. Simplified 52.8

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

      [Start] 52.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 52.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 52.8	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 53.4	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 52.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 52.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 52.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 52.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 26.4

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

   4. Simplified 21.7

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

      [Start] 26.4	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 28.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 28.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 28.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 28.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 21.7	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 11.3

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

   6. Simplified 12.3

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

      [Start] 11.3	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      times-frac [=>] 5.3	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]
      associate-/r/ [=>] 5.3	\[ \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}} \]
      associate-*l/ [=>] 5.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{-t}{\color{blue}{\frac{\cos k \cdot \left(-\ell\right)}{k}}}} \]
      distribute-rgt-neg-out [=>] 5.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{-t}{\frac{\color{blue}{-\cos k \cdot \ell}}{k}}} \]
      distribute-lft-neg-out [<=] 5.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{-t}{\frac{\color{blue}{\left(-\cos k\right) \cdot \ell}}{k}}} \]
      associate-/r/ [=>] 12.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \color{blue}{\left(\frac{-t}{\left(-\cos k\right) \cdot \ell} \cdot k\right)}} \]
      *-commutative [=>] 12.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(\frac{-t}{\color{blue}{\ell \cdot \left(-\cos k\right)}} \cdot k\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-41} \lor \neg \left(t \leq 1.05 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{t}{\ell}} \cdot \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \frac{-t}{\ell \cdot \left(-\cos k\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	20873

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-26} \lor \neg \left(t \leq 1.8 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\tan k}}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \frac{-t}{\ell \cdot \left(-\cos k\right)}\right)}\\ \end{array} \]

Alternative 2
Error	7.9
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \left(t \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ t_3 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{if}\;k \leq -2.2 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+111}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{t_1} \cdot \frac{\frac{\cos k}{k \cdot k}}{t}\right)\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	7.4
Cost	20748

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ t_2 := 2 + \frac{k \cdot k}{t \cdot t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+160}:\\ \;\;\;\;t_1 \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-36}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \frac{-t}{\ell \cdot \left(-\cos k\right)}\right)}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+150}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 4
Error	8.0
Cost	20620

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ t_2 := 2 + \frac{k \cdot k}{t \cdot t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+160}:\\ \;\;\;\;t_1 \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-41}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+146}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 5
Error	7.8
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{+72} \lor \neg \left(k \leq 5.6 \cdot 10^{+88}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \left(t \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \end{array} \]

Alternative 6
Error	11.8
Cost	14928

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ t_2 := t_1 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+160}:\\ \;\;\;\;t_1 \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{1 - \cos \left(k + k\right)}{\ell \cdot \frac{2 \cdot \ell}{t}}}\\ \mathbf{elif}\;t \leq 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 7
Error	11.9
Cost	14928

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ t_2 := 2 + \frac{k \cdot k}{t \cdot t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+160}:\\ \;\;\;\;t_1 \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-68}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{1 - \cos \left(k + k\right)}{\ell \cdot \frac{2 \cdot \ell}{t}}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+146}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 8
Error	15.9
Cost	14736

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -0.98:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-264}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{1 - \cos \left(k + k\right)}{\ell \cdot \frac{2 \cdot \ell}{t}}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\right) \cdot \frac{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{t \cdot k}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.8
Cost	14672

\[\begin{array}{l} t_1 := \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{1 - \cos \left(k + k\right)}{\ell \cdot \frac{2 \cdot \ell}{t}}}\\ t_2 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -13:\\ \;\;\;\;t_2 \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.05 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 116000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 10
Error	16.8
Cost	14540

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{t}}{0.5 + \cos \left(k + k\right) \cdot -0.5}\\ \mathbf{if}\;k \leq -62000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 11
Error	17.0
Cost	14153

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-6} \lor \neg \left(t \leq 38000\right):\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 12
Error	17.0
Cost	14152

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -0.00025:\\ \;\;\;\;t_1 \cdot \frac{\frac{0.5}{\sin k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 34000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{t \cdot \sin k}\right)\\ \end{array} \]

Alternative 13
Error	18.5
Cost	13960

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\frac{0.5}{k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 14
Error	18.7
Cost	7752

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\frac{0.5}{k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 15
Error	18.7
Cost	7620

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \left(0.5 \cdot t_1\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 16
Error	18.8
Cost	7620

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \left(0.5 \cdot \frac{\frac{\ell}{t}}{k}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 17
Error	18.7
Cost	7620

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}} \cdot \frac{\frac{0.5}{k}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 18
Error	19.2
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -8.7 \cdot 10^{-47} \lor \neg \left(t \leq 1.3 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 19
Error	20.0
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-57} \lor \neg \left(t \leq 1.02 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 20
Error	29.0
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-6} \lor \neg \left(t \leq 7.8 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 21
Error	27.3
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-30} \lor \neg \left(t \leq 10^{-83}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{\left(t \cdot t\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 22
Error	22.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-7} \lor \neg \left(t \leq -7 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\ell}}\\ \end{array} \]

Alternative 23
Error	29.2
Cost	832

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

Alternative 24
Error	27.9
Cost	832

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

Reproduce

herbie shell --seed 2023073 
(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))))