Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 0.8

Time: 31.9s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -6.6 \cdot 10^{-56} \lor \neg \left(k \leq 1.25 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\frac{\frac{\ell}{k}}{t}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

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
 (let* ((t_1 (* k (/ k l))))
   (if (or (<= k -6.6e-56) (not (<= k 1.25e-11)))
     (/ 2.0 (/ (/ (pow (sin k) 2.0) (/ (/ l k) t)) (* l (/ (cos k) k))))
     (* 2.0 (/ 1.0 (* t_1 (* t t_1)))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((k <= -6.6e-56) || !(k <= 1.25e-11)) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / ((l / k) / t)) / (l * (cos(k) / k)));
	} else {
		tmp = 2.0 * (1.0 / (t_1 * (t * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k / l)
    if ((k <= (-6.6d-56)) .or. (.not. (k <= 1.25d-11))) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / ((l / k) / t)) / (l * (cos(k) / k)))
    else
        tmp = 2.0d0 * (1.0d0 / (t_1 * (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((k <= -6.6e-56) || !(k <= 1.25e-11)) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / ((l / k) / t)) / (l * (Math.cos(k) / k)));
	} else {
		tmp = 2.0 * (1.0 / (t_1 * (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

↓

def code(t, l, k):
	t_1 = k * (k / l)
	tmp = 0
	if (k <= -6.6e-56) or not (k <= 1.25e-11):
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / ((l / k) / t)) / (l * (math.cos(k) / k)))
	else:
		tmp = 2.0 * (1.0 / (t_1 * (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

↓

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if ((k <= -6.6e-56) || !(k <= 1.25e-11))
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / Float64(Float64(l / k) / t)) / Float64(l * Float64(cos(k) / k))));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64(t_1 * Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

↓

function tmp_2 = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 0.0;
	if ((k <= -6.6e-56) || ~((k <= 1.25e-11)))
		tmp = 2.0 / (((sin(k) ^ 2.0) / ((l / k) / t)) / (l * (cos(k) / k)));
	else
		tmp = 2.0 * (1.0 / (t_1 * (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -6.6e-56], N[Not[LessEqual[k, 1.25e-11]], $MachinePrecision]], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(t$95$1 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -6.59999999999999967e-56 or 1.25000000000000005e-11 < k

   1. Initial program 44.8

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

   2. Simplified 36.7

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 44.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 44.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 44.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 44.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 36.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 36.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 19.6

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

   4. Simplified 15.3

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

      [Start] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.5	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 19.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 15.3	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Taylor expanded in k around inf 19.6

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

   6. Simplified 4.6

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

      [Start] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      *-commutative [<=] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.5	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      associate-*l/ [<=] 19.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell \cdot \ell} \cdot {\sin k}^{2}\right)}} \]
      associate-*l/ [=>] 16.8	\[ \frac{2}{\color{blue}{\frac{k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot {\sin k}^{2}\right)}{\frac{\cos k}{k}}}} \]
      associate-*r* [=>] 16.8	\[ \frac{2}{\frac{\color{blue}{\left(k \cdot \frac{t}{\ell \cdot \ell}\right) \cdot {\sin k}^{2}}}{\frac{\cos k}{k}}} \]
      associate-/r* [=>] 11.6	\[ \frac{2}{\frac{\left(k \cdot \color{blue}{\frac{\frac{t}{\ell}}{\ell}}\right) \cdot {\sin k}^{2}}{\frac{\cos k}{k}}} \]
      associate-*r/ [=>] 5.9	\[ \frac{2}{\frac{\color{blue}{\frac{k \cdot \frac{t}{\ell}}{\ell}} \cdot {\sin k}^{2}}{\frac{\cos k}{k}}} \]
      associate-/r/ [<=] 5.8	\[ \frac{2}{\frac{\color{blue}{\frac{k \cdot \frac{t}{\ell}}{\frac{\ell}{{\sin k}^{2}}}}}{\frac{\cos k}{k}}} \]
      associate-/r* [<=] 4.5	\[ \frac{2}{\color{blue}{\frac{k \cdot \frac{t}{\ell}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}}} \]
      associate-*l/ [=>] 4.5	\[ \frac{2}{\frac{k \cdot \frac{t}{\ell}}{\color{blue}{\frac{\ell \cdot \frac{\cos k}{k}}{{\sin k}^{2}}}}} \]
      *-commutative [<=] 4.5	\[ \frac{2}{\frac{k \cdot \frac{t}{\ell}}{\frac{\color{blue}{\frac{\cos k}{k} \cdot \ell}}{{\sin k}^{2}}}} \]

   7. Taylor expanded in k around 0 4.6

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

   8. Applied egg-rr 0.7

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

   if -6.59999999999999967e-56 < k < 1.25000000000000005e-11

   1. Initial program 62.5

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

   2. Simplified 50.2

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

      [Start] 62.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 62.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 62.4	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 62.4	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 61.4	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 61.4	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 61.4	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 61.7	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 61.7	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 61.7	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 45.6

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 42.1

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]
      Proof

      [Start] 45.6	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 45.6	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 42.1	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 18.4

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

   6. Taylor expanded in t around 0 18.4

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

   7. Simplified 18.3

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

      [Start] 18.4	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \]
      *-commutative [=>] 18.4	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{2}}}\right) \]
      unpow2 [=>] 18.4	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot k\right)}}\right) \]
      associate-*r* [=>] 18.3	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot k}}\right) \]
      *-commutative [=>] 18.3	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot k\right)}}\right) \]
      *-commutative [=>] 18.3	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}}\right) \]

   8. Applied egg-rr 0.9

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.6 \cdot 10^{-56} \lor \neg \left(k \leq 1.25 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\frac{\frac{\ell}{k}}{t}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.4
Cost	20620

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \ell \cdot \cos k\\ t_3 := 2 \cdot \frac{t_2}{\frac{t}{\frac{\ell}{k}} \cdot \left(k \cdot t_1\right)}\\ t_4 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{-107}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{-240}:\\ \;\;\;\;2 \cdot \frac{1}{t_4 \cdot \left(t \cdot t_4\right)}\\ \mathbf{elif}\;\ell \leq 7.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)} \cdot \frac{t_2}{k}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	3.9
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4 \cdot 10^{-54} \lor \neg \left(k \leq 4.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \frac{\frac{k \cdot t}{\ell}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	3.8
Cost	20356

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.85 \cdot 10^{-57}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \cos k}{\frac{t}{\frac{\ell}{k}} \cdot \left(k \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot t}{\ell}}{\ell \cdot \frac{\cos k}{k}} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ \end{array} \]

Alternative 4
Error	4.5
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2900000 \lor \neg \left(k \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\left(0.5 - \frac{\cos \left(k + k\right)}{2}\right) \cdot \frac{k \cdot \frac{t}{\ell}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	4.4
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2900000 \lor \neg \left(k \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot t}{\ell}}{\ell \cdot \frac{\cos k}{k}} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 6
Error	13.1
Cost	14408

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\ell}\right)}\\ \end{array} \]

Alternative 7
Error	13.1
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.85 \cdot 10^{-30} \lor \neg \left(k \leq 2.3 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Error	22.8
Cost	1088

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ 2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Alternative 9
Error	25.7
Cost	960

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

Alternative 10
Error	23.2
Cost	960

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

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))))