Toniolo and Linder, Equation (10+)

Average Accuracy: 49.0% → 88.8%

Time: 34.3s

Precision: binary64

Cost: 21000

Math

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

↓

\[\begin{array}{l} t_1 := \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{t_1}}}\\ \end{array} \]

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
         (* (* t (* (/ t l) (sin k))) (* (tan k) (+ 2.0 (pow (/ k t) 2.0))))))
   (if (<= t -2e-101)
     (/ 2.0 (/ t_1 (/ l t)))
     (if (<= t 1.4e-85)
       (/ 2.0 (* (/ (* t (- k)) (/ (- l) (/ k l))) (* (sin k) (tan k))))
       (/ 2.0 (/ 1.0 (/ (/ l t) t_1)))))))

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 = (t * ((t / l) * sin(k))) * (tan(k) * (2.0 + pow((k / t), 2.0)));
	double tmp;
	if (t <= -2e-101) {
		tmp = 2.0 / (t_1 / (l / t));
	} else if (t <= 1.4e-85) {
		tmp = 2.0 / (((t * -k) / (-l / (k / l))) * (sin(k) * tan(k)));
	} else {
		tmp = 2.0 / (1.0 / ((l / t) / t_1));
	}
	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 = (t * ((t / l) * sin(k))) * (tan(k) * (2.0d0 + ((k / t) ** 2.0d0)))
    if (t <= (-2d-101)) then
        tmp = 2.0d0 / (t_1 / (l / t))
    else if (t <= 1.4d-85) then
        tmp = 2.0d0 / (((t * -k) / (-l / (k / l))) * (sin(k) * tan(k)))
    else
        tmp = 2.0d0 / (1.0d0 / ((l / t) / t_1))
    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 = (t * ((t / l) * Math.sin(k))) * (Math.tan(k) * (2.0 + Math.pow((k / t), 2.0)));
	double tmp;
	if (t <= -2e-101) {
		tmp = 2.0 / (t_1 / (l / t));
	} else if (t <= 1.4e-85) {
		tmp = 2.0 / (((t * -k) / (-l / (k / l))) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = 2.0 / (1.0 / ((l / t) / t_1));
	}
	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 = (t * ((t / l) * math.sin(k))) * (math.tan(k) * (2.0 + math.pow((k / t), 2.0)))
	tmp = 0
	if t <= -2e-101:
		tmp = 2.0 / (t_1 / (l / t))
	elif t <= 1.4e-85:
		tmp = 2.0 / (((t * -k) / (-l / (k / l))) * (math.sin(k) * math.tan(k)))
	else:
		tmp = 2.0 / (1.0 / ((l / t) / t_1))
	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(t * Float64(Float64(t / l) * sin(k))) * Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))
	tmp = 0.0
	if (t <= -2e-101)
		tmp = Float64(2.0 / Float64(t_1 / Float64(l / t)));
	elseif (t <= 1.4e-85)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(-k)) / Float64(Float64(-l) / Float64(k / l))) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(1.0 / Float64(Float64(l / t) / t_1)));
	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 = (t * ((t / l) * sin(k))) * (tan(k) * (2.0 + ((k / t) ^ 2.0)));
	tmp = 0.0;
	if (t <= -2e-101)
		tmp = 2.0 / (t_1 / (l / t));
	elseif (t <= 1.4e-85)
		tmp = 2.0 / (((t * -k) / (-l / (k / l))) * (sin(k) * tan(k)));
	else
		tmp = 2.0 / (1.0 / ((l / t) / t_1));
	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[(t * N[(N[(t / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e-101], N[(2.0 / N[(t$95$1 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-85], N[(2.0 / N[(N[(N[(t * (-k)), $MachinePrecision] / N[((-l) / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(1.0 / N[(N[(l / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.0000000000000001e-101

   1. Initial program 63.5%

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

   2. Simplified 63.5%

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

      [Start] 63.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 63.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 63.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 73.1%

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

      [Start] 63.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 63.5	\[ \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 73.1	\[ \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 86.4%

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

      [Start] 73.1	\[ \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l* [=>] 76.8	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/l* [=>] 84.4	\[ \frac{2}{\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l/ [=>] 86.4	\[ \frac{2}{\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 90.6%

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

      [Start] 86.4	\[ \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l/ [=>] 90.6	\[ \frac{2}{\color{blue}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{t}}}} \]
      associate-+r+ [=>] 90.6	\[ \frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{\frac{\ell}{t}}} \]
      metadata-eval [=>] 90.6	\[ \frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{\ell}{t}}} \]

   if -2.0000000000000001e-101 < t < 1.40000000000000008e-85

   1. Initial program 3.0%

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

   2. Simplified 3.2%

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

      [Start] 3.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 3.0	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 3.0	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 3.2	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 3.2	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 3.2	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 3.2	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around inf 59.7%

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

   4. Simplified 56.0%

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

      [Start] 59.7	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-/l* [=>] 56.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 56.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{t}} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 56.0	\[ \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{t}} \cdot \left(\sin k \cdot \tan k\right)} \]

   5. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(-k \cdot k\right) \cdot \frac{1}{\frac{\ell \cdot \left(-\ell\right)}{t}}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      Proof

      [Start] 56.0	\[ \frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{t}} \cdot \left(\sin k \cdot \tan k\right)} \]
      frac-2neg [=>] 56.0	\[ \frac{2}{\color{blue}{\frac{-k \cdot k}{-\frac{\ell \cdot \ell}{t}}} \cdot \left(\sin k \cdot \tan k\right)} \]
      div-inv [=>] 56.0	\[ \frac{2}{\color{blue}{\left(\left(-k \cdot k\right) \cdot \frac{1}{-\frac{\ell \cdot \ell}{t}}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      distribute-neg-frac [=>] 56.0	\[ \frac{2}{\left(\left(-k \cdot k\right) \cdot \frac{1}{\color{blue}{\frac{-\ell \cdot \ell}{t}}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      distribute-rgt-neg-in [=>] 56.0	\[ \frac{2}{\left(\left(-k \cdot k\right) \cdot \frac{1}{\frac{\color{blue}{\ell \cdot \left(-\ell\right)}}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 81.7%

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

      [Start] 56.0	\[ \frac{2}{\left(\left(-k \cdot k\right) \cdot \frac{1}{\frac{\ell \cdot \left(-\ell\right)}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-*r/ [=>] 56.0	\[ \frac{2}{\color{blue}{\frac{\left(-k \cdot k\right) \cdot 1}{\frac{\ell \cdot \left(-\ell\right)}{t}}} \cdot \left(\sin k \cdot \tan k\right)} \]
      *-rgt-identity [=>] 56.0	\[ \frac{2}{\frac{\color{blue}{-k \cdot k}}{\frac{\ell \cdot \left(-\ell\right)}{t}} \cdot \left(\sin k \cdot \tan k\right)} \]
      distribute-neg-frac [<=] 56.0	\[ \frac{2}{\color{blue}{\left(-\frac{k \cdot k}{\frac{\ell \cdot \left(-\ell\right)}{t}}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-/r/ [=>] 55.4	\[ \frac{2}{\left(-\color{blue}{\frac{k \cdot k}{\ell \cdot \left(-\ell\right)} \cdot t}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      distribute-rgt-neg-out [<=] 55.4	\[ \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell \cdot \left(-\ell\right)} \cdot \left(-t\right)\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-/l* [=>] 61.0	\[ \frac{2}{\left(\color{blue}{\frac{k}{\frac{\ell \cdot \left(-\ell\right)}{k}}} \cdot \left(-t\right)\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-*l/ [=>] 71.4	\[ \frac{2}{\color{blue}{\frac{k \cdot \left(-t\right)}{\frac{\ell \cdot \left(-\ell\right)}{k}}} \cdot \left(\sin k \cdot \tan k\right)} \]
      *-commutative [=>] 71.4	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\color{blue}{\left(-\ell\right) \cdot \ell}}{k}} \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-/l* [=>] 81.7	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\color{blue}{\frac{-\ell}{\frac{k}{\ell}}}} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 1.40000000000000008e-85 < t

   1. Initial program 63.2%

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

   2. Simplified 63.3%

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

      [Start] 63.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 63.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 63.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 73.4%

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

      [Start] 63.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 63.3	\[ \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 73.4	\[ \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 86.9%

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

      [Start] 73.4	\[ \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l* [=>] 77.5	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/l* [=>] 85.0	\[ \frac{2}{\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l/ [=>] 86.9	\[ \frac{2}{\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 91.4%

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

      [Start] 86.9	\[ \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l/ [=>] 91.4	\[ \frac{2}{\color{blue}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\frac{\ell}{t}}}} \]
      clear-num [=>] 91.4	\[ \frac{2}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}}} \]
      associate-+r+ [=>] 91.4	\[ \frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}}} \]
      metadata-eval [=>] 91.4	\[ \frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.8%
Cost	20873

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-101} \lor \neg \left(t \leq 3.7 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 2
Accuracy	89.1%
Cost	20873

\[\begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-86} \lor \neg \left(t \leq 6.8 \cdot 10^{-85}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(t \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 3
Accuracy	88.8%
Cost	20873

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-101} \lor \neg \left(t \leq 6.5 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 4
Accuracy	84.2%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;k \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\left(t \cdot \tan k\right) \cdot {\left(\frac{k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 5
Accuracy	81.6%
Cost	14600

\[\begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{t \cdot k}{\ell}}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot t_1}\\ \end{array} \]

Alternative 6
Accuracy	84.2%
Cost	14600

\[\begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot t_1}\\ \end{array} \]

Alternative 7
Accuracy	79.9%
Cost	14152

\[\begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(-k\right)}{\frac{-\ell}{\frac{k}{\ell}}} \cdot t_1}\\ \end{array} \]

Alternative 8
Accuracy	79.3%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -2.25 \cdot 10^{-19} \lor \neg \left(k \leq 2.35 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 9
Accuracy	73.0%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;t \leq -0.000125:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-79}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{-2}\\ \end{array} \]

Alternative 10
Accuracy	73.8%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{-2}\\ \end{array} \]

Alternative 11
Accuracy	66.6%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-79}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{-2}\\ \end{array} \]

Alternative 12
Accuracy	68.3%
Cost	7436

\[\begin{array}{l} t_1 := \frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{2 \cdot \frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{\ell}}}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-79}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+66}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	68.3%
Cost	7436

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-79}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Accuracy	68.0%
Cost	7304

\[\begin{array}{l} t_1 := \frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{2 \cdot \frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{\ell}}}}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Accuracy	60.8%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+46} \lor \neg \left(t \leq 7.4 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{2 \cdot \frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{t \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 16
Accuracy	54.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-29} \lor \neg \left(t \leq 12000000\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 17
Accuracy	54.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+23} \lor \neg \left(t \leq 6.8 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t \cdot t}\\ \end{array} \]

Alternative 18
Accuracy	54.2%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+25} \lor \neg \left(t \leq 5.9 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t \cdot t}\\ \end{array} \]

Alternative 19
Accuracy	54.6%
Cost	1097

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot t}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+43} \lor \neg \left(t \leq 5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot t_1}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 20
Accuracy	54.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;k \leq -1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t \cdot t}}{t \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t \cdot t}\\ \end{array} \]

Alternative 21
Accuracy	53.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 22
Accuracy	53.6%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 23
Accuracy	45.2%
Cost	832

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

Alternative 24
Accuracy	45.1%
Cost	832

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

Reproduce

herbie shell --seed 2023151 
(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))))