Toniolo and Linder, Equation (10-)

Average Error: 47.2 → 12.8

Time: 33.1s

Precision: binary64

Cost: 20360

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{{k}^{2}}\\ \mathbf{if}\;k \leq -2.9 \cdot 10^{-66}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2}}\right)\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;t_1 \cdot \frac{1}{t \cdot \left(0.5 \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\frac{t_1}{t}}{\tan k}\right)\\ \end{array} \]

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
 (let* ((t_1 (/ l (pow k 2.0))))
   (if (<= k -2.9e-66)
     (* l (* (/ l t) (/ (/ 2.0 (* (sin k) (tan k))) (pow k 2.0))))
     (if (<= k 3.8e-69)
       (* t_1 (/ 1.0 (* t (* 0.5 (/ (pow k 2.0) l)))))
       (* l (* (/ 2.0 (sin k)) (/ (/ t_1 t) (tan 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 t_1 = l / pow(k, 2.0);
	double tmp;
	if (k <= -2.9e-66) {
		tmp = l * ((l / t) * ((2.0 / (sin(k) * tan(k))) / pow(k, 2.0)));
	} else if (k <= 3.8e-69) {
		tmp = t_1 * (1.0 / (t * (0.5 * (pow(k, 2.0) / l))));
	} else {
		tmp = l * ((2.0 / sin(k)) * ((t_1 / t) / tan(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) :: t_1
    real(8) :: tmp
    t_1 = l / (k ** 2.0d0)
    if (k <= (-2.9d-66)) then
        tmp = l * ((l / t) * ((2.0d0 / (sin(k) * tan(k))) / (k ** 2.0d0)))
    else if (k <= 3.8d-69) then
        tmp = t_1 * (1.0d0 / (t * (0.5d0 * ((k ** 2.0d0) / l))))
    else
        tmp = l * ((2.0d0 / sin(k)) * ((t_1 / t) / tan(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 t_1 = l / Math.pow(k, 2.0);
	double tmp;
	if (k <= -2.9e-66) {
		tmp = l * ((l / t) * ((2.0 / (Math.sin(k) * Math.tan(k))) / Math.pow(k, 2.0)));
	} else if (k <= 3.8e-69) {
		tmp = t_1 * (1.0 / (t * (0.5 * (Math.pow(k, 2.0) / l))));
	} else {
		tmp = l * ((2.0 / Math.sin(k)) * ((t_1 / t) / Math.tan(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):
	t_1 = l / math.pow(k, 2.0)
	tmp = 0
	if k <= -2.9e-66:
		tmp = l * ((l / t) * ((2.0 / (math.sin(k) * math.tan(k))) / math.pow(k, 2.0)))
	elif k <= 3.8e-69:
		tmp = t_1 * (1.0 / (t * (0.5 * (math.pow(k, 2.0) / l))))
	else:
		tmp = l * ((2.0 / math.sin(k)) * ((t_1 / t) / math.tan(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)
	t_1 = Float64(l / (k ^ 2.0))
	tmp = 0.0
	if (k <= -2.9e-66)
		tmp = Float64(l * Float64(Float64(l / t) * Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / (k ^ 2.0))));
	elseif (k <= 3.8e-69)
		tmp = Float64(t_1 * Float64(1.0 / Float64(t * Float64(0.5 * Float64((k ^ 2.0) / l)))));
	else
		tmp = Float64(l * Float64(Float64(2.0 / sin(k)) * Float64(Float64(t_1 / t) / tan(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)
	t_1 = l / (k ^ 2.0);
	tmp = 0.0;
	if (k <= -2.9e-66)
		tmp = l * ((l / t) * ((2.0 / (sin(k) * tan(k))) / (k ^ 2.0)));
	elseif (k <= 3.8e-69)
		tmp = t_1 * (1.0 / (t * (0.5 * ((k ^ 2.0) / l))));
	else
		tmp = l * ((2.0 / sin(k)) * ((t_1 / t) / tan(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_] := Block[{t$95$1 = N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -2.9e-66], N[(l * N[(N[(l / t), $MachinePrecision] * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e-69], N[(t$95$1 * N[(1.0 / N[(t * N[(0.5 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / t), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.90000000000000011e-66

   1. Initial program 45.1

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

   2. Simplified 36.8

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

      [Start] 45.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-46 [=>] 45.1	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-48 [=>] 36.8	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 36.8	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-4 [=>] 36.8	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 35.0

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

   4. Simplified 33.8

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

      [Start] 35.0	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 35.0	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 35.0	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 35.0	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-46 [=>] 35.0	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-61 [=>] 34.9	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [=>] 33.8	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
      rational.json-simplify-47 [=>] 33.8	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   5. Taylor expanded in l around 0 15.3

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

   6. Applied egg-rr 12.6

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   7. Applied egg-rr 12.8

      \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{t}}{{k}^{2}} + 0} \]

   8. Simplified 12.8

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

      [Start] 12.8	\[ \ell \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{t}}{{k}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 12.8	\[ \color{blue}{\ell \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{t}}{{k}^{2}}} \]
      rational.json-simplify-49 [=>] 12.8	\[ \ell \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\frac{\frac{2}{\sin k}}{\tan k}}{{k}^{2}}\right)} \]
      rational.json-simplify-47 [=>] 12.8	\[ \ell \cdot \left(\frac{\ell}{t} \cdot \frac{\color{blue}{\frac{2}{\sin k \cdot \tan k}}}{{k}^{2}}\right) \]

   if -2.90000000000000011e-66 < k < 3.7999999999999998e-69

   1. Initial program 63.8

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

   2. Simplified 60.6

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

      [Start] 63.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-46 [=>] 63.8	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-48 [=>] 60.6	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 60.6	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-4 [=>] 60.6	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 58.5

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

   4. Simplified 53.7

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

      [Start] 58.5	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 58.5	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 58.4	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 58.4	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-46 [=>] 55.5	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-61 [=>] 54.0	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [=>] 53.7	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
      rational.json-simplify-47 [=>] 53.7	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   5. Taylor expanded in l around 0 27.7

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

   6. Applied egg-rr 19.2

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   7. Taylor expanded in k around 0 19.3

      \[\leadsto \frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \color{blue}{\left(0.5 \cdot \frac{{k}^{2}}{\ell}\right)}} \]

   if 3.7999999999999998e-69 < k

   1. Initial program 45.2

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

   2. Simplified 37.2

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

      [Start] 45.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-46 [=>] 45.3	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-48 [=>] 37.2	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 37.2	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-4 [=>] 37.2	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 35.2

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

   4. Simplified 33.9

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

      [Start] 35.2	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 35.2	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 35.2	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 35.2	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-46 [=>] 35.1	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-61 [=>] 35.0	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [=>] 33.9	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
      rational.json-simplify-47 [=>] 33.9	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   5. Taylor expanded in l around 0 13.4

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

   6. Applied egg-rr 11.7

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

   7. Simplified 11.2

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

      [Start] 11.7	\[ \ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{{k}^{2} \cdot \tan k}\right) + 0 \]
      rational.json-simplify-4 [=>] 11.7	\[ \color{blue}{\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{{k}^{2} \cdot \tan k}\right)} \]
      rational.json-simplify-46 [=>] 11.7	\[ \ell \cdot \left(\frac{2}{\sin k} \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{{k}^{2}}}{\tan k}}\right) \]
      rational.json-simplify-44 [=>] 11.2	\[ \ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{{k}^{2}}}{t}}}{\tan k}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.9 \cdot 10^{-66}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{2}{\sin k \cdot \tan k}}{{k}^{2}}\right)\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \left(0.5 \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\tan k}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	14.8
Cost	20360

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -5.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \left(0.5 \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.6
Cost	20360

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \frac{\frac{\frac{\frac{2}{\tan k}}{{k}^{2}}}{t}}{\sin k}\right)\\ \mathbf{if}\;k \leq -5.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \left(0.5 \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.0
Cost	20096

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

Alternative 4
Error	12.6
Cost	20096

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

Alternative 5
Error	25.0
Cost	13632

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

Alternative 6
Error	30.2
Cost	7040

\[\ell \cdot \left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \]

Alternative 7
Error	29.7
Cost	7040

\[\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \]

Reproduce

herbie shell --seed 2023075 
(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))))