Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 12.8

Time: 32.7s

Precision: binary64

Cost: 20744

Math

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

↓

\[\begin{array}{l} t_1 := \frac{2}{\sin k}\\ t_2 := \frac{\tan k}{\ell \cdot t_1}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\ell \cdot \left(t_1 \cdot \frac{\ell}{{k}^{2} \cdot \left(\tan k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(t \cdot \left({k}^{2} \cdot t_2\right)\right)} \cdot \left(\ell + \ell\right)\\ \end{array} \]

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 (/ 2.0 (sin k))) (t_2 (/ (tan k) (* l t_1))))
   (if (<= k -1.35e+130)
     (* l (* t_1 (/ l (* (pow k 2.0) (* (tan k) t)))))
     (if (<= k 8.5e+95)
       (* (/ l (pow k 2.0)) (/ 1.0 (* t t_2)))
       (* (/ 1.0 (* 2.0 (* t (* (pow k 2.0) t_2)))) (+ l 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 t_1 = 2.0 / sin(k);
	double t_2 = tan(k) / (l * t_1);
	double tmp;
	if (k <= -1.35e+130) {
		tmp = l * (t_1 * (l / (pow(k, 2.0) * (tan(k) * t))));
	} else if (k <= 8.5e+95) {
		tmp = (l / pow(k, 2.0)) * (1.0 / (t * t_2));
	} else {
		tmp = (1.0 / (2.0 * (t * (pow(k, 2.0) * t_2)))) * (l + 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 / sin(k)
    t_2 = tan(k) / (l * t_1)
    if (k <= (-1.35d+130)) then
        tmp = l * (t_1 * (l / ((k ** 2.0d0) * (tan(k) * t))))
    else if (k <= 8.5d+95) then
        tmp = (l / (k ** 2.0d0)) * (1.0d0 / (t * t_2))
    else
        tmp = (1.0d0 / (2.0d0 * (t * ((k ** 2.0d0) * t_2)))) * (l + 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 t_1 = 2.0 / Math.sin(k);
	double t_2 = Math.tan(k) / (l * t_1);
	double tmp;
	if (k <= -1.35e+130) {
		tmp = l * (t_1 * (l / (Math.pow(k, 2.0) * (Math.tan(k) * t))));
	} else if (k <= 8.5e+95) {
		tmp = (l / Math.pow(k, 2.0)) * (1.0 / (t * t_2));
	} else {
		tmp = (1.0 / (2.0 * (t * (Math.pow(k, 2.0) * t_2)))) * (l + 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):
	t_1 = 2.0 / math.sin(k)
	t_2 = math.tan(k) / (l * t_1)
	tmp = 0
	if k <= -1.35e+130:
		tmp = l * (t_1 * (l / (math.pow(k, 2.0) * (math.tan(k) * t))))
	elif k <= 8.5e+95:
		tmp = (l / math.pow(k, 2.0)) * (1.0 / (t * t_2))
	else:
		tmp = (1.0 / (2.0 * (t * (math.pow(k, 2.0) * t_2)))) * (l + 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)
	t_1 = Float64(2.0 / sin(k))
	t_2 = Float64(tan(k) / Float64(l * t_1))
	tmp = 0.0
	if (k <= -1.35e+130)
		tmp = Float64(l * Float64(t_1 * Float64(l / Float64((k ^ 2.0) * Float64(tan(k) * t)))));
	elseif (k <= 8.5e+95)
		tmp = Float64(Float64(l / (k ^ 2.0)) * Float64(1.0 / Float64(t * t_2)));
	else
		tmp = Float64(Float64(1.0 / Float64(2.0 * Float64(t * Float64((k ^ 2.0) * t_2)))) * Float64(l + 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)
	t_1 = 2.0 / sin(k);
	t_2 = tan(k) / (l * t_1);
	tmp = 0.0;
	if (k <= -1.35e+130)
		tmp = l * (t_1 * (l / ((k ^ 2.0) * (tan(k) * t))));
	elseif (k <= 8.5e+95)
		tmp = (l / (k ^ 2.0)) * (1.0 / (t * t_2));
	else
		tmp = (1.0 / (2.0 * (t * ((k ^ 2.0) * t_2)))) * (l + 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_] := Block[{t$95$1 = N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] / N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.35e+130], N[(l * N[(t$95$1 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.5e+95], N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(2.0 * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.3499999999999999e130

   1. Initial program 40.6

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

   2. Simplified 34.9

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

      [Start] 40.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-46 [=>] 40.7	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-48 [=>] 34.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 34.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-4 [=>] 34.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 32.4

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

   4. Simplified 32.1

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

      [Start] 32.4	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 32.4	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 32.4	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 32.4	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-46 [=>] 32.4	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-61 [=>] 32.4	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [=>] 32.1	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
      rational.json-simplify-47 [=>] 32.1	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   5. Taylor expanded in l around 0 22.2

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

   6. Applied egg-rr 21.7

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

   7. Simplified 22.2

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

      [Start] 21.7	\[ \ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{{k}^{2} \cdot \tan k}\right) + 0 \]
      rational.json-simplify-4 [=>] 21.7	\[ \color{blue}{\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{{k}^{2} \cdot \tan k}\right)} \]
      rational.json-simplify-47 [=>] 22.3	\[ \ell \cdot \left(\frac{2}{\sin k} \cdot \color{blue}{\frac{\ell}{t \cdot \left({k}^{2} \cdot \tan k\right)}}\right) \]
      rational.json-simplify-43 [=>] 22.2	\[ \ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot \left(\tan k \cdot t\right)}}\right) \]

   if -1.3499999999999999e130 < k < 8.5000000000000002e95

   1. Initial program 55.2

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

   2. Simplified 46.1

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

      [Start] 55.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-46 [=>] 55.3	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-48 [=>] 46.1	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 46.1	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-4 [=>] 46.1	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 44.1

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

   4. Simplified 41.4

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

      [Start] 44.1	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 44.1	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 44.1	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 44.1	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-46 [=>] 43.4	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-61 [=>] 42.8	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [=>] 41.4	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
      rational.json-simplify-47 [=>] 41.4	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   5. Taylor expanded in l around 0 10.0

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

   6. Applied egg-rr 5.0

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

   if 8.5000000000000002e95 < k

   1. Initial program 40.7

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

   2. Simplified 33.9

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

      [Start] 40.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-46 [=>] 40.8	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-48 [=>] 33.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 33.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-4 [=>] 33.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 31.8

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

   4. Simplified 31.0

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

      [Start] 31.8	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 31.8	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 31.8	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-46 [=>] 31.8	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-46 [=>] 31.8	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-61 [=>] 31.8	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [=>] 31.0	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
      rational.json-simplify-47 [=>] 31.0	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

   5. Taylor expanded in l around 0 19.2

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

   6. Applied egg-rr 17.6

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\ell}{{k}^{2} \cdot \left(\tan k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{1}{t \cdot \frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(t \cdot \left({k}^{2} \cdot \frac{\tan k}{\ell \cdot \frac{2}{\sin k}}\right)\right)} \cdot \left(\ell + \ell\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{{k}^{2}}\\ t_2 := \frac{2}{\sin k}\\ t_3 := \frac{\tan k}{\ell \cdot t_2}\\ \mathbf{if}\;k \leq -1.7 \cdot 10^{+131}:\\ \;\;\;\;\ell \cdot \left(t_2 \cdot \frac{\ell}{{k}^{2} \cdot \left(\tan k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+104}:\\ \;\;\;\;t_1 \cdot \frac{1}{t \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t}}{t_3}\\ \end{array} \]

Alternative 2
Error	14.4
Cost	20360

\[\begin{array}{l} t_1 := \ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\ell}{{k}^{2} \cdot \left(\tan k \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{0.5 \cdot \frac{{k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	12.7
Cost	20360

\[\begin{array}{l} t_1 := \left(\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{{k}^{2} \cdot \tan k}\right) \cdot \ell\\ \mathbf{if}\;k \leq -4 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{0.5 \cdot \frac{{k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.5
Cost	20360

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{{k}^{2}}}{t}\\ t_2 := \frac{2}{\sin k} \cdot \left(\ell \cdot \frac{t_1}{\tan k}\right)\\ \mathbf{if}\;k \leq -2.65 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_1}{0.5 \cdot \frac{{k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	12.5
Cost	20096

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

Alternative 6
Error	25.4
Cost	13632

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

Alternative 7
Error	24.2
Cost	13632

\[\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{0.5 \cdot \frac{{k}^{2}}{\ell}} \]

Alternative 8
Error	31.5
Cost	13376

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

Alternative 9
Error	31.4
Cost	13376

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

Alternative 10
Error	31.4
Cost	13376

\[\frac{\frac{{\ell}^{2}}{\frac{{k}^{4}}{2}}}{t} \]

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))))