Toniolo and Linder, Equation (10+)

Average Accuracy: 49.1% → 87.2%

Time: 39.3s

Precision: binary64

Cost: 21004

Math

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

↓

\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -9 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.32 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (/ (* (tan k) (* t (* k k))) (/ l (sin k))) l)))
        (t_2
         (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))
   (if (<= k -9e+151)
     t_2
     (if (<= k -1.32e+47)
       t_1
       (if (<= k -8e-58)
         (*
          (/ (/ 2.0 (* (tan k) (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))) t)
          (* (/ l t) (/ l t)))
         (if (<= k 1.0)
           (* (/ (/ l t) (* k t)) (/ l (* k t)))
           (if (<= k 2.4e+131) t_1 t_2)))))))

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 / (((tan(k) * (t * (k * k))) / (l / sin(k))) / l);
	double t_2 = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	double tmp;
	if (k <= -9e+151) {
		tmp = t_2;
	} else if (k <= -1.32e+47) {
		tmp = t_1;
	} else if (k <= -8e-58) {
		tmp = ((2.0 / (tan(k) * (sin(k) * (2.0 + pow((k / t), 2.0))))) / t) * ((l / t) * (l / t));
	} else if (k <= 1.0) {
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	} else if (k <= 2.4e+131) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 / (((tan(k) * (t * (k * k))) / (l / sin(k))) / l)
    t_2 = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    if (k <= (-9d+151)) then
        tmp = t_2
    else if (k <= (-1.32d+47)) then
        tmp = t_1
    else if (k <= (-8d-58)) then
        tmp = ((2.0d0 / (tan(k) * (sin(k) * (2.0d0 + ((k / t) ** 2.0d0))))) / t) * ((l / t) * (l / t))
    else if (k <= 1.0d0) then
        tmp = ((l / t) / (k * t)) * (l / (k * t))
    else if (k <= 2.4d+131) then
        tmp = t_1
    else
        tmp = t_2
    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.tan(k) * (t * (k * k))) / (l / Math.sin(k))) / l);
	double t_2 = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	double tmp;
	if (k <= -9e+151) {
		tmp = t_2;
	} else if (k <= -1.32e+47) {
		tmp = t_1;
	} else if (k <= -8e-58) {
		tmp = ((2.0 / (Math.tan(k) * (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0))))) / t) * ((l / t) * (l / t));
	} else if (k <= 1.0) {
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	} else if (k <= 2.4e+131) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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.tan(k) * (t * (k * k))) / (l / math.sin(k))) / l)
	t_2 = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	tmp = 0
	if k <= -9e+151:
		tmp = t_2
	elif k <= -1.32e+47:
		tmp = t_1
	elif k <= -8e-58:
		tmp = ((2.0 / (math.tan(k) * (math.sin(k) * (2.0 + math.pow((k / t), 2.0))))) / t) * ((l / t) * (l / t))
	elif k <= 1.0:
		tmp = ((l / t) / (k * t)) * (l / (k * t))
	elif k <= 2.4e+131:
		tmp = t_1
	else:
		tmp = t_2
	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 / Float64(Float64(Float64(tan(k) * Float64(t * Float64(k * k))) / Float64(l / sin(k))) / l))
	t_2 = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))))
	tmp = 0.0
	if (k <= -9e+151)
		tmp = t_2;
	elseif (k <= -1.32e+47)
		tmp = t_1;
	elseif (k <= -8e-58)
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) / t) * Float64(Float64(l / t) * Float64(l / t)));
	elseif (k <= 1.0)
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * Float64(l / Float64(k * t)));
	elseif (k <= 2.4e+131)
		tmp = t_1;
	else
		tmp = t_2;
	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 / (((tan(k) * (t * (k * k))) / (l / sin(k))) / l);
	t_2 = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	tmp = 0.0;
	if (k <= -9e+151)
		tmp = t_2;
	elseif (k <= -1.32e+47)
		tmp = t_1;
	elseif (k <= -8e-58)
		tmp = ((2.0 / (tan(k) * (sin(k) * (2.0 + ((k / t) ^ 2.0))))) / t) * ((l / t) * (l / t));
	elseif (k <= 1.0)
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	elseif (k <= 2.4e+131)
		tmp = t_1;
	else
		tmp = t_2;
	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[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -9e+151], t$95$2, If[LessEqual[k, -1.32e+47], t$95$1, If[LessEqual[k, -8e-58], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.4e+131], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -8.9999999999999997e151 or 2.3999999999999999e131 < k

   1. Initial program 46.6%

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

   2. Simplified 46.6%

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

      [Start] 46.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 46.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 46.6	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 46.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 46.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 46.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 46.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 46.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 62.3%

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

   4. Simplified 90.6%

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

      [Start] 62.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 62.3	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 62.1	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 62.1	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 62.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 90.6	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 90.6	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -8.9999999999999997e151 < k < -1.31999999999999992e47 or 1 < k < 2.3999999999999999e131

   1. Initial program 53.6%

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

   2. Simplified 53.5%

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

      [Start] 53.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 53.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 53.6	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 53.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 53.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 53.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 53.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 53.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 60.0%

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

   4. Taylor expanded in t around 0 84.3%

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

   5. Simplified 84.3%

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

      [Start] 84.3	\[ \frac{2}{\frac{\frac{\tan k \cdot \left({k}^{2} \cdot t\right)}{\frac{\ell}{\sin k}}}{\ell}} \]
      unpow2 [=>] 84.3	\[ \frac{2}{\frac{\frac{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}{\frac{\ell}{\sin k}}}{\ell}} \]

   if -1.31999999999999992e47 < k < -8.0000000000000002e-58

   1. Initial program 54.0%

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

   2. Simplified 54.2%

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

      [Start] 54.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 54.0	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 54.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 54.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 54.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 54.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 54.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 54.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 54.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 69.8%

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

   4. Applied egg-rr 80.8%

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

   if -8.0000000000000002e-58 < k < 1

   1. Initial program 47.6%

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

   2. Simplified 53.3%

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

      [Start] 47.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 47.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 48.1	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 47.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 53.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 53.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 53.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 53.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 30.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Simplified 32.7%

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

      [Start] 30.5	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 30.5	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 32.7	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 32.7	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 37.7%

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

   6. Taylor expanded in k around 0 32.7%

      \[\leadsto \frac{\ell}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]

   7. Simplified 70.6%

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

      [Start] 32.7	\[ \frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]
      unpow2 [=>] 32.7	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      unpow3 [=>] 32.7	\[ \frac{\ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{\ell}} \]
      associate-*r* [=>] 35.6	\[ \frac{\ell}{\frac{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}}{\ell}} \]
      swap-sqr [<=] 64.8	\[ \frac{\ell}{\frac{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t}{\ell}} \]
      unpow2 [<=] 64.8	\[ \frac{\ell}{\frac{\color{blue}{{\left(k \cdot t\right)}^{2}} \cdot t}{\ell}} \]
      *-commutative [=>] 64.8	\[ \frac{\ell}{\frac{\color{blue}{t \cdot {\left(k \cdot t\right)}^{2}}}{\ell}} \]
      *-commutative [<=] 64.8	\[ \frac{\ell}{\frac{\color{blue}{{\left(k \cdot t\right)}^{2} \cdot t}}{\ell}} \]
      associate-*l/ [<=] 70.6	\[ \frac{\ell}{\color{blue}{\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot t}} \]
      *-commutative [=>] 70.6	\[ \frac{\ell}{\color{blue}{t \cdot \frac{{\left(k \cdot t\right)}^{2}}{\ell}}} \]

   8. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -1.32 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ \mathbf{elif}\;k \leq -8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.1%
Cost	21004

\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -4.4 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.8 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.35 \cdot 10^{-100}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t \cdot \frac{t}{\ell}}}{t \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 0.0037:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	87.1%
Cost	20884

\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.22 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.45 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left({k}^{4}, \frac{t \cdot \left(t \cdot 0.3333333333333333\right)}{\ell} + \frac{1}{\ell}, 2 \cdot \frac{t \cdot t}{\frac{\frac{\ell}{k}}{k}}\right)}\\ \mathbf{elif}\;k \leq 0.0044:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	78.0%
Cost	14984

\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ \mathbf{if}\;k \leq -11200000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left({k}^{4}, \frac{t \cdot \left(t \cdot 0.3333333333333333\right)}{\ell} + \frac{1}{\ell}, 2 \cdot \frac{t \cdot t}{\frac{\frac{\ell}{k}}{k}}\right)}\\ \mathbf{elif}\;k \leq 0.0027:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	76.7%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -4.8 \cdot 10^{-43} \lor \neg \left(k \leq 0.004\right):\\ \;\;\;\;\frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{\sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 5
Accuracy	69.9%
Cost	9097

\[\begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{-118} \lor \neg \left(k \leq 2.6 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left({k}^{4} \cdot \left(2 \cdot \frac{t \cdot -0.16666666666666666 + t \cdot 0.3333333333333333}{\ell} + \frac{1}{\ell \cdot t}\right) + 2 \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 6
Accuracy	68.8%
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_1}{\left(k \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot t_1\\ \end{array} \]

Alternative 7
Accuracy	65.5%
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{t_1}{\left(k \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 10^{-124}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot t_1\\ \end{array} \]

Alternative 8
Accuracy	65.9%
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{t_1}{\left(k \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2}{2 \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot t_1\\ \end{array} \]

Alternative 9
Accuracy	65.5%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{-29} \lor \neg \left(t \leq 8.8 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{t}\\ \end{array} \]

Alternative 10
Accuracy	65.5%
Cost	1096

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{t_1}{\left(k \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot t_1\\ \end{array} \]

Alternative 11
Accuracy	53.1%
Cost	832

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

Reproduce

herbie shell --seed 2023129 
(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))))