Toniolo and Linder, Equation (10-)

Average Accuracy: 26.2% → 99.4%

Time: 52.9s

Precision: binary64

Cost: 14089

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;k \leq -1.12 \cdot 10^{-63} \lor \neg \left(k \leq 3 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\frac{-2}{\tan k}}{\frac{-k}{\ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_1 \cdot \frac{2}{t}\right)\\ \end{array} \]

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 k) k)))
   (if (or (<= k -1.12e-63) (not (<= k 3e-85)))
     (* (/ l (* k (sin k))) (/ (/ -2.0 (tan k)) (* (/ (- k) l) t)))
     (* t_1 (* t_1 (/ 2.0 t))))))

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 / k) / k;
	double tmp;
	if ((k <= -1.12e-63) || !(k <= 3e-85)) {
		tmp = (l / (k * sin(k))) * ((-2.0 / tan(k)) / ((-k / l) * t));
	} else {
		tmp = t_1 * (t_1 * (2.0 / t));
	}
	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) / k
    if ((k <= (-1.12d-63)) .or. (.not. (k <= 3d-85))) then
        tmp = (l / (k * sin(k))) * (((-2.0d0) / tan(k)) / ((-k / l) * t))
    else
        tmp = t_1 * (t_1 * (2.0d0 / t))
    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 / k) / k;
	double tmp;
	if ((k <= -1.12e-63) || !(k <= 3e-85)) {
		tmp = (l / (k * Math.sin(k))) * ((-2.0 / Math.tan(k)) / ((-k / l) * t));
	} else {
		tmp = t_1 * (t_1 * (2.0 / t));
	}
	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 / k) / k
	tmp = 0
	if (k <= -1.12e-63) or not (k <= 3e-85):
		tmp = (l / (k * math.sin(k))) * ((-2.0 / math.tan(k)) / ((-k / l) * t))
	else:
		tmp = t_1 * (t_1 * (2.0 / t))
	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(Float64(l / k) / k)
	tmp = 0.0
	if ((k <= -1.12e-63) || !(k <= 3e-85))
		tmp = Float64(Float64(l / Float64(k * sin(k))) * Float64(Float64(-2.0 / tan(k)) / Float64(Float64(Float64(-k) / l) * t)));
	else
		tmp = Float64(t_1 * Float64(t_1 * Float64(2.0 / t)));
	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) / k;
	tmp = 0.0;
	if ((k <= -1.12e-63) || ~((k <= 3e-85)))
		tmp = (l / (k * sin(k))) * ((-2.0 / tan(k)) / ((-k / l) * t));
	else
		tmp = t_1 * (t_1 * (2.0 / t));
	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[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]}, If[Or[LessEqual[k, -1.12e-63], N[Not[LessEqual[k, 3e-85]], $MachinePrecision]], N[(N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[((-k) / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$1 * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -1.12000000000000002e-63 or 3.00000000000000022e-85 < k

   1. Initial program 28.9%

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

   2. Simplified 42.0%

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

      [Start] 28.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 28.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 28.9	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 28.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 42.0	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 42.0	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 69.5%

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

   4. Simplified 80.3%

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

      [Start] 69.5	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 69.5	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 69.5	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 80.3	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 80.3	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 80.3	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 93.1%

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

      [Start] 80.3	\[ \frac{2}{\tan k \cdot \left(\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-/r* [=>] 80.3	\[ \color{blue}{\frac{\frac{2}{\tan k}}{\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}}} \]
      associate-/l/ [<=] 80.6	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{\frac{k \cdot \left(k \cdot \sin k\right)}{\ell}}} \]
      associate-/l* [=>] 87.2	\[ \frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{\color{blue}{\frac{k}{\frac{\ell}{k \cdot \sin k}}}} \]
      frac-2neg [=>] 87.2	\[ \frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{\color{blue}{\frac{-k}{-\frac{\ell}{k \cdot \sin k}}}} \]
      associate-/r/ [=>] 93.0	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right)} \]
      clear-num [=>] 93.0	\[ \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{1}{\frac{\ell}{t}}}}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      associate-/r/ [=>] 93.1	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{1} \cdot \frac{\ell}{t}}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      associate-/l/ [=>] 93.1	\[ \frac{\color{blue}{\frac{2}{1 \cdot \tan k}} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      *-un-lft-identity [<=] 93.1	\[ \frac{\frac{2}{\color{blue}{\tan k}} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      associate-/r* [=>] 93.1	\[ \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\color{blue}{\frac{\frac{\ell}{k}}{\sin k}}\right) \]

   6. Simplified 99.4%

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

      [Start] 93.1	\[ \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\frac{\frac{\ell}{k}}{\sin k}\right) \]
      *-commutative [=>] 93.1	\[ \color{blue}{\left(-\frac{\frac{\ell}{k}}{\sin k}\right) \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k}} \]
      associate-/l/ [=>] 93.1	\[ \left(-\color{blue}{\frac{\ell}{\sin k \cdot k}}\right) \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \]
      *-commutative [<=] 93.1	\[ \left(-\frac{\ell}{\color{blue}{k \cdot \sin k}}\right) \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \]
      associate-/l* [=>] 93.2	\[ \left(-\frac{\ell}{k \cdot \sin k}\right) \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\frac{-k}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 99.4	\[ \left(-\frac{\ell}{k \cdot \sin k}\right) \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\frac{-k}{\ell} \cdot t}} \]

   if -1.12000000000000002e-63 < k < 3.00000000000000022e-85

   1. Initial program 0.8%

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

   2. Simplified 5.2%

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

      [Start] 0.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 0.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 0.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 5.2	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 5.2	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 28.6%

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

   4. Simplified 24.7%

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

      [Start] 28.6	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 16.4	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 16.4	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 24.7	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 24.7	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 24.7	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 63.5%

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

      [Start] 24.7	\[ \frac{2}{\tan k \cdot \left(\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-/r* [=>] 24.7	\[ \color{blue}{\frac{\frac{2}{\tan k}}{\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}}} \]
      associate-/l/ [<=] 24.8	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{\frac{k \cdot \left(k \cdot \sin k\right)}{\ell}}} \]
      associate-/l* [=>] 49.1	\[ \frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{\color{blue}{\frac{k}{\frac{\ell}{k \cdot \sin k}}}} \]
      frac-2neg [=>] 49.1	\[ \frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{\color{blue}{\frac{-k}{-\frac{\ell}{k \cdot \sin k}}}} \]
      associate-/r/ [=>] 54.3	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\ell}}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right)} \]
      clear-num [=>] 54.3	\[ \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{1}{\frac{\ell}{t}}}}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      associate-/r/ [=>] 55.9	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{1} \cdot \frac{\ell}{t}}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      associate-/l/ [=>] 55.9	\[ \frac{\color{blue}{\frac{2}{1 \cdot \tan k}} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      *-un-lft-identity [<=] 55.9	\[ \frac{\frac{2}{\color{blue}{\tan k}} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\frac{\ell}{k \cdot \sin k}\right) \]
      associate-/r* [=>] 63.5	\[ \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\color{blue}{\frac{\frac{\ell}{k}}{\sin k}}\right) \]

   6. Simplified 68.3%

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

      [Start] 63.5	\[ \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \cdot \left(-\frac{\frac{\ell}{k}}{\sin k}\right) \]
      *-commutative [=>] 63.5	\[ \color{blue}{\left(-\frac{\frac{\ell}{k}}{\sin k}\right) \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k}} \]
      associate-/l/ [=>] 55.9	\[ \left(-\color{blue}{\frac{\ell}{\sin k \cdot k}}\right) \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \]
      *-commutative [<=] 55.9	\[ \left(-\frac{\ell}{\color{blue}{k \cdot \sin k}}\right) \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{-k} \]
      associate-/l* [=>] 55.9	\[ \left(-\frac{\ell}{k \cdot \sin k}\right) \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\frac{-k}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 68.3	\[ \left(-\frac{\ell}{k \cdot \sin k}\right) \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\frac{-k}{\ell} \cdot t}} \]

   7. Taylor expanded in k around 0 68.3%

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

   8. Simplified 90.2%

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

      [Start] 68.3	\[ \left(-\frac{\ell}{{k}^{2}}\right) \cdot \frac{\frac{2}{\tan k}}{\frac{-k}{\ell} \cdot t} \]
      unpow2 [=>] 68.3	\[ \left(-\frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\frac{2}{\tan k}}{\frac{-k}{\ell} \cdot t} \]
      associate-/r* [=>] 90.2	\[ \left(-\color{blue}{\frac{\frac{\ell}{k}}{k}}\right) \cdot \frac{\frac{2}{\tan k}}{\frac{-k}{\ell} \cdot t} \]

   9. Taylor expanded in k around 0 59.5%

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

   10. Simplified 98.9%

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

       [Start] 59.5	\[ \left(-\frac{\frac{\ell}{k}}{k}\right) \cdot \left(-2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \]
       associate-*r/ [=>] 59.5	\[ \left(-\frac{\frac{\ell}{k}}{k}\right) \cdot \color{blue}{\frac{-2 \cdot \ell}{{k}^{2} \cdot t}} \]
       *-commutative [=>] 59.5	\[ \left(-\frac{\frac{\ell}{k}}{k}\right) \cdot \frac{-2 \cdot \ell}{\color{blue}{t \cdot {k}^{2}}} \]
       times-frac [=>] 73.7	\[ \left(-\frac{\frac{\ell}{k}}{k}\right) \cdot \color{blue}{\left(\frac{-2}{t} \cdot \frac{\ell}{{k}^{2}}\right)} \]
       unpow2 [=>] 73.7	\[ \left(-\frac{\frac{\ell}{k}}{k}\right) \cdot \left(\frac{-2}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]
       associate-/r* [=>] 98.9	\[ \left(-\frac{\frac{\ell}{k}}{k}\right) \cdot \left(\frac{-2}{t} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.12 \cdot 10^{-63} \lor \neg \left(k \leq 3 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\frac{-2}{\tan k}}{\frac{-k}{\ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{2}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.7%
Cost	14025

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;k \leq -2 \cdot 10^{-33} \lor \neg \left(k \leq 1.8 \cdot 10^{-34}\right):\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{k}{\frac{\ell}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_1 \cdot \frac{2}{t}\right)\\ \end{array} \]

Alternative 2
Accuracy	92.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -5.2 \cdot 10^{-79} \lor \neg \left(k \leq 1.1 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(t \cdot 0.5\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	95.3%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-97} \lor \neg \left(t \leq 7.2 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\frac{2}{\tan k} \cdot \frac{\ell}{k \cdot t}\right)\\ \end{array} \]

Alternative 4
Accuracy	94.1%
Cost	14024

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{\tan k \cdot \left(k \cdot \frac{t}{\ell}\right)}}{\sin k}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\\ \end{array} \]

Alternative 5
Accuracy	93.6%
Cost	14024

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\\ \end{array} \]

Alternative 6
Accuracy	59.4%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+150} \lor \neg \left(t \leq -1.36 \cdot 10^{-306}\right):\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 7
Accuracy	59.5%
Cost	960

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

Alternative 8
Accuracy	59.5%
Cost	960

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

Alternative 9
Accuracy	64.6%
Cost	960

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

Reproduce

herbie shell --seed 2023133 
(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))))