Toniolo and Linder, Equation (10+)

Average Accuracy: 54.6% → 79.3%

Time: 38.3s

Precision: binary64

Cost: 33801

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-58} \lor \neg \left(t \leq 5.4 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\left(\frac{\frac{\ell}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{\tan k}{\sqrt{2}}}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \end{array} \]

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 8.2e-58) (not (<= t 5.4e-46)))
   (/
    (*
     (* (/ (/ l t) (+ 2.0 (pow (/ k t) 2.0))) (/ (sqrt 2.0) (sin k)))
     (/ (/ l t) (/ (tan k) (sqrt 2.0))))
    t)
   (* 2.0 (/ (/ (cos k) (* (/ k l) (* t (pow (sin k) 2.0)))) (/ k 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 tmp;
	if ((t <= 8.2e-58) || !(t <= 5.4e-46)) {
		tmp = ((((l / t) / (2.0 + pow((k / t), 2.0))) * (sqrt(2.0) / sin(k))) * ((l / t) / (tan(k) / sqrt(2.0)))) / t;
	} else {
		tmp = 2.0 * ((cos(k) / ((k / l) * (t * pow(sin(k), 2.0)))) / (k / 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) :: tmp
    if ((t <= 8.2d-58) .or. (.not. (t <= 5.4d-46))) then
        tmp = ((((l / t) / (2.0d0 + ((k / t) ** 2.0d0))) * (sqrt(2.0d0) / sin(k))) * ((l / t) / (tan(k) / sqrt(2.0d0)))) / t
    else
        tmp = 2.0d0 * ((cos(k) / ((k / l) * (t * (sin(k) ** 2.0d0)))) / (k / 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 tmp;
	if ((t <= 8.2e-58) || !(t <= 5.4e-46)) {
		tmp = ((((l / t) / (2.0 + Math.pow((k / t), 2.0))) * (Math.sqrt(2.0) / Math.sin(k))) * ((l / t) / (Math.tan(k) / Math.sqrt(2.0)))) / t;
	} else {
		tmp = 2.0 * ((Math.cos(k) / ((k / l) * (t * Math.pow(Math.sin(k), 2.0)))) / (k / 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):
	tmp = 0
	if (t <= 8.2e-58) or not (t <= 5.4e-46):
		tmp = ((((l / t) / (2.0 + math.pow((k / t), 2.0))) * (math.sqrt(2.0) / math.sin(k))) * ((l / t) / (math.tan(k) / math.sqrt(2.0)))) / t
	else:
		tmp = 2.0 * ((math.cos(k) / ((k / l) * (t * math.pow(math.sin(k), 2.0)))) / (k / 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)
	tmp = 0.0
	if ((t <= 8.2e-58) || !(t <= 5.4e-46))
		tmp = Float64(Float64(Float64(Float64(Float64(l / t) / Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(sqrt(2.0) / sin(k))) * Float64(Float64(l / t) / Float64(tan(k) / sqrt(2.0)))) / t);
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(Float64(k / l) * Float64(t * (sin(k) ^ 2.0)))) / Float64(k / 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)
	tmp = 0.0;
	if ((t <= 8.2e-58) || ~((t <= 5.4e-46)))
		tmp = ((((l / t) / (2.0 + ((k / t) ^ 2.0))) * (sqrt(2.0) / sin(k))) * ((l / t) / (tan(k) / sqrt(2.0)))) / t;
	else
		tmp = 2.0 * ((cos(k) / ((k / l) * (t * (sin(k) ^ 2.0)))) / (k / 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_] := If[Or[LessEqual[t, 8.2e-58], N[Not[LessEqual[t, 5.4e-46]], $MachinePrecision]], N[(N[(N[(N[(N[(l / t), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.20000000000000056e-58 or 5.4e-46 < t

   1. Initial program 53.9%

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

   2. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]
      Step-by-step derivation

      [Start] 53.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 53.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/l/ [<=] 54.0	\[ \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]
      *-commutative [=>] 54.0	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 55.9	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]
      associate-/l* [=>] 54.4	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
      associate-/r/ [=>] 50.2	\[ \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Applied egg-rr 54.4%

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

      [Start] 57.4	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right) \]
      associate-*l/ [=>] 50.2	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}} \]
      cube-mult [=>] 50.2	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot t\right)}} \]
      associate-/r* [=>] 54.4	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot t}} \]

   4. Applied egg-rr 68.5%

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

      [Start] 54.4	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\frac{\ell \cdot \ell}{t}}{t \cdot t} \]
      *-commutative [=>] 54.4	\[ \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot t} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      associate-/r* [=>] 58.5	\[ \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t}}{t}}{t}} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-*l/ [=>] 59.4	\[ \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t}}{t} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{t}} \]
      associate-/l* [=>] 66.6	\[ \frac{\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{t} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{t} \]
      associate-/l/ [=>] 68.5	\[ \frac{\color{blue}{\frac{\ell}{t \cdot \frac{t}{\ell}}} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{t} \]

   5. Applied egg-rr 75.2%

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

      [Start] 68.5	\[ \frac{\frac{\ell}{t \cdot \frac{t}{\ell}} \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{t} \]
      associate-*r/ [=>] 68.9	\[ \frac{\color{blue}{\frac{\frac{\ell}{t \cdot \frac{t}{\ell}} \cdot 2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{t} \]
      associate-*r* [=>] 68.9	\[ \frac{\frac{\frac{\ell}{t \cdot \frac{t}{\ell}} \cdot 2}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}}}{t} \]
      associate-/r* [=>] 74.0	\[ \frac{\color{blue}{\frac{\frac{\frac{\ell}{t \cdot \frac{t}{\ell}} \cdot 2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{\tan k}}}{t} \]

   6. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k} \cdot \frac{\frac{\ell}{t} \cdot \sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}}{t} \]
      Step-by-step derivation

      [Start] 75.2	\[ \frac{\frac{\frac{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{\tan k}}{t} \]
      associate-/l/ [=>] 69.4	\[ \frac{\color{blue}{\frac{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}}{t} \]
      add-sqr-sqrt [=>] 69.4	\[ \frac{\frac{\color{blue}{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2} \cdot \sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t} \]
      times-frac [=>] 75.1	\[ \frac{\color{blue}{\frac{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}{\tan k} \cdot \frac{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}}{t} \]
      sqrt-prod [=>] 75.1	\[ \frac{\frac{\color{blue}{\sqrt{{\left(\frac{\ell}{t}\right)}^{2}} \cdot \sqrt{2}}}{\tan k} \cdot \frac{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      unpow2 [=>] 75.1	\[ \frac{\frac{\sqrt{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} \cdot \sqrt{2}}{\tan k} \cdot \frac{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      sqrt-prod [=>] 41.5	\[ \frac{\frac{\color{blue}{\left(\sqrt{\frac{\ell}{t}} \cdot \sqrt{\frac{\ell}{t}}\right)} \cdot \sqrt{2}}{\tan k} \cdot \frac{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      add-sqr-sqrt [<=] 55.5	\[ \frac{\frac{\color{blue}{\frac{\ell}{t}} \cdot \sqrt{2}}{\tan k} \cdot \frac{\sqrt{{\left(\frac{\ell}{t}\right)}^{2} \cdot 2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      sqrt-prod [=>] 55.5	\[ \frac{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k} \cdot \frac{\color{blue}{\sqrt{{\left(\frac{\ell}{t}\right)}^{2}} \cdot \sqrt{2}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      unpow2 [=>] 55.5	\[ \frac{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k} \cdot \frac{\sqrt{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} \cdot \sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      sqrt-prod [=>] 45.0	\[ \frac{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k} \cdot \frac{\color{blue}{\left(\sqrt{\frac{\ell}{t}} \cdot \sqrt{\frac{\ell}{t}}\right)} \cdot \sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      add-sqr-sqrt [<=] 81.0	\[ \frac{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{t}} \cdot \sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]

   7. Simplified 81.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{\ell}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{\tan k}{\sqrt{2}}}}}{t} \]
      Step-by-step derivation

      [Start] 81.0	\[ \frac{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k} \cdot \frac{\frac{\ell}{t} \cdot \sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{t} \]
      *-commutative [=>] 81.0	\[ \frac{\color{blue}{\frac{\frac{\ell}{t} \cdot \sqrt{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k}}}{t} \]
      times-frac [=>] 81.1	\[ \frac{\color{blue}{\left(\frac{\frac{\ell}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sin k}\right)} \cdot \frac{\frac{\ell}{t} \cdot \sqrt{2}}{\tan k}}{t} \]
      associate-/l* [=>] 81.1	\[ \frac{\left(\frac{\frac{\ell}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{\tan k}{\sqrt{2}}}}}{t} \]

   if 8.20000000000000056e-58 < t < 5.4e-46

   1. Initial program 33.3%

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

   2. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{2}{\left({t}^{3} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\ell \cdot \ell\right)} \]
      Step-by-step derivation

      [Start] 33.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 33.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l/ [=>] 33.3	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right)} \]
      associate-*l/ [=>] 33.3	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 33.3	\[ \frac{2}{\color{blue}{\frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right)}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 33.3	\[ \color{blue}{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)} \]

   3. Taylor expanded in t around 0 34.8%

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

   4. Simplified 99.5%

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

      [Start] 34.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 34.8	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 34.8	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 34.8	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 34.8	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 99.5	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. Applied egg-rr 99.0%

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

      [Start] 99.5	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      associate-*l* [=>] 99.5	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\right)} \]
      clear-num [=>] 99.5	\[ 2 \cdot \left(\color{blue}{\frac{1}{\frac{k}{\ell}}} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\right) \]
      associate-*l/ [=>] 99.5	\[ 2 \cdot \color{blue}{\frac{1 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)}{\frac{k}{\ell}}} \]
      *-un-lft-identity [<=] 99.5	\[ 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}}}{\frac{k}{\ell}} \]
      clear-num [=>] 99.5	\[ 2 \cdot \frac{\color{blue}{\frac{1}{\frac{k}{\ell}}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}}{\frac{k}{\ell}} \]
      frac-times [=>] 99.0	\[ 2 \cdot \frac{\color{blue}{\frac{1 \cdot \cos k}{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}}}{\frac{k}{\ell}} \]
      *-un-lft-identity [<=] 99.0	\[ 2 \cdot \frac{\frac{\color{blue}{\cos k}}{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}}{\frac{k}{\ell}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-58} \lor \neg \left(t \leq 5.4 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\left(\frac{\frac{\ell}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\frac{\tan k}{\sqrt{2}}}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	67.9%
Cost	27476

\[\begin{array}{l} t_1 := 2 \cdot {\left(\frac{\ell}{t}\right)}^{2}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{t_3 \cdot \tan k}\\ \mathbf{if}\;t \leq 5.6 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-57}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+78}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\frac{t_1}{t_3 \cdot \sin k}}{\tan k}}{t}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t_1}{2 \cdot \sin k}}{\tan k}}{t}\\ \end{array} \]

Alternative 2
Accuracy	69.0%
Cost	27472

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 1.16 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{t_1 \cdot \tan k}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{k}{\frac{\ell}{k}} \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t}\right)}^{2}}{\frac{t_1}{\frac{2}{\sin k}}} \cdot \frac{1}{t \cdot \tan k}\\ \end{array} \]

Alternative 3
Accuracy	68.1%
Cost	27344

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \ell \cdot \frac{\ell \cdot \frac{2}{\tan k}}{t_2 \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq 2.7 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\frac{\ell}{t}}{t}\right) \cdot \frac{2}{t_2 \cdot \left(\sin k \cdot \tan k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	68.1%
Cost	27344

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 1.65 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{t_2 \cdot \tan k}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{\tan k}}{t_2 \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\frac{\ell}{t}}{t}\right) \cdot \frac{2}{t_2 \cdot \left(\sin k \cdot \tan k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	71.2%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-10} \lor \neg \left(k \leq 7.5 \cdot 10^{-6}\right):\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{k}}{k \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	71.2%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-10} \lor \neg \left(k \leq 0.000145\right):\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	71.7%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-10} \lor \neg \left(k \leq 2.4 \cdot 10^{-21}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+96} \lor \neg \left(t \leq 2.4 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	68.0%
Cost	14288

\[\begin{array}{l} t_1 := \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(\sin k \cdot \left(t \cdot k\right)\right)\right)}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-249}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	67.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq 9.1 \cdot 10^{+95} \lor \neg \left(t \leq 3 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \end{array} \]

Alternative 11
Accuracy	68.0%
Cost	7436

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-46}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	64.9%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \end{array} \]

Alternative 13
Accuracy	64.9%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \frac{t}{\ell}} \cdot \frac{2}{2 \cdot \left(k \cdot k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \end{array} \]

Alternative 14
Accuracy	65.2%
Cost	960

\[\frac{\ell}{k} \cdot \left(\frac{1}{t} \cdot \frac{\frac{\ell}{k}}{t \cdot t}\right) \]

Alternative 15
Accuracy	57.9%
Cost	832

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

Alternative 16
Accuracy	57.8%
Cost	832

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

Alternative 17
Accuracy	64.3%
Cost	832

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

Reproduce

herbie shell --seed 2023159 
(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))))