Toniolo and Linder, Equation (10-)

Average Error: 74.1% → 1.22%

Time: 38.8s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \frac{\frac{k}{\cos k}}{\ell \cdot {\sin k}^{-2}}\right)}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}}\\ \end{array} \]

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 (* k (/ k l))))
   (if (<= k -1.8e-15)
     (/ 2.0 (* (/ k l) (* t (/ (/ k (cos k)) (* l (pow (sin k) -2.0))))))
     (if (<= k 3.4e-9)
       (/ 2.0 (* t_1 (* t t_1)))
       (/ 2.0 (/ (* (/ k l) t) (* (/ l (pow (sin k) 2.0)) (/ (cos k) k))))))))

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 = k * (k / l);
	double tmp;
	if (k <= -1.8e-15) {
		tmp = 2.0 / ((k / l) * (t * ((k / cos(k)) / (l * pow(sin(k), -2.0)))));
	} else if (k <= 3.4e-9) {
		tmp = 2.0 / (t_1 * (t * t_1));
	} else {
		tmp = 2.0 / (((k / l) * t) / ((l / pow(sin(k), 2.0)) * (cos(k) / k)));
	}
	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 = k * (k / l)
    if (k <= (-1.8d-15)) then
        tmp = 2.0d0 / ((k / l) * (t * ((k / cos(k)) / (l * (sin(k) ** (-2.0d0))))))
    else if (k <= 3.4d-9) then
        tmp = 2.0d0 / (t_1 * (t * t_1))
    else
        tmp = 2.0d0 / (((k / l) * t) / ((l / (sin(k) ** 2.0d0)) * (cos(k) / k)))
    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 = k * (k / l);
	double tmp;
	if (k <= -1.8e-15) {
		tmp = 2.0 / ((k / l) * (t * ((k / Math.cos(k)) / (l * Math.pow(Math.sin(k), -2.0)))));
	} else if (k <= 3.4e-9) {
		tmp = 2.0 / (t_1 * (t * t_1));
	} else {
		tmp = 2.0 / (((k / l) * t) / ((l / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / k)));
	}
	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 = k * (k / l)
	tmp = 0
	if k <= -1.8e-15:
		tmp = 2.0 / ((k / l) * (t * ((k / math.cos(k)) / (l * math.pow(math.sin(k), -2.0)))))
	elif k <= 3.4e-9:
		tmp = 2.0 / (t_1 * (t * t_1))
	else:
		tmp = 2.0 / (((k / l) * t) / ((l / math.pow(math.sin(k), 2.0)) * (math.cos(k) / k)))
	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(k * Float64(k / l))
	tmp = 0.0
	if (k <= -1.8e-15)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(t * Float64(Float64(k / cos(k)) / Float64(l * (sin(k) ^ -2.0))))));
	elseif (k <= 3.4e-9)
		tmp = Float64(2.0 / Float64(t_1 * Float64(t * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * t) / Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(cos(k) / k))));
	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 = k * (k / l);
	tmp = 0.0;
	if (k <= -1.8e-15)
		tmp = 2.0 / ((k / l) * (t * ((k / cos(k)) / (l * (sin(k) ^ -2.0)))));
	elseif (k <= 3.4e-9)
		tmp = 2.0 / (t_1 * (t * t_1));
	else
		tmp = 2.0 / (((k / l) * t) / ((l / (sin(k) ^ 2.0)) * (cos(k) / k)));
	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[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.8e-15], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(t * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.4e-9], N[(2.0 / N[(t$95$1 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.8000000000000001e-15

   1. Initial program 68.95

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

   2. Simplified 68.95

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

      [Start] 68.95	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 68.95	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 68.95	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 56.71	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 56.71	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]
      metadata-eval [<=] 56.71	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + -1\right)}\right)\right)} \]
      metadata-eval [<=] 56.71	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \color{blue}{\left(-1\right)}\right)\right)\right)} \]
      associate-+l+ [<=] 68.95	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(-1\right)\right)}\right)} \]
      +-commutative [<=] 68.95	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + \left(-1\right)\right)\right)} \]
      metadata-eval [=>] 68.95	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \color{blue}{-1}\right)\right)} \]

   3. Taylor expanded in t around 0 29.88

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

   4. Simplified 23.71

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

      [Start] 29.88	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 29.82	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 29.82	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 29.82	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 29.82	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 29.82	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 23.71	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 6.85

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

   6. Taylor expanded in k around 0 7

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

   7. Simplified 1.14

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

      [Start] 7	\[ \frac{2}{\frac{\frac{k \cdot t}{\ell}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}} \]
      associate-*l/ [<=] 1.14	\[ \frac{2}{\frac{\color{blue}{\frac{k}{\ell} \cdot t}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}} \]

   8. Applied egg-rr 1.15

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

   if -1.8000000000000001e-15 < k < 3.3999999999999998e-9

   1. Initial program 96.57

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

   2. Simplified 77.08

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

      [Start] 96.57	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 96.66	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 96.66	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 97.54	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 94.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 94.44	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 94.44	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 77.08	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 77.08	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 77.08	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around 0 65.96

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

   4. Simplified 66.52

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{4}}} \]
      Proof

      [Start] 65.96	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      associate-*r/ [=>] 65.96	\[ \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]
      *-commutative [=>] 65.96	\[ \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]
      times-frac [=>] 66.52	\[ \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
      unpow2 [=>] 66.52	\[ \frac{2}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}} \]

   5. Applied egg-rr 26.64

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

   6. Applied egg-rr 1.38

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

   if 3.3999999999999998e-9 < k

   1. Initial program 68.63

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

   2. Simplified 68.63

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

      [Start] 68.63	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 68.63	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 68.63	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 56.57	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 56.57	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]
      metadata-eval [<=] 56.57	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + -1\right)}\right)\right)} \]
      metadata-eval [<=] 56.57	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \color{blue}{\left(-1\right)}\right)\right)\right)} \]
      associate-+l+ [<=] 68.63	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + \left(-1\right)\right)}\right)} \]
      +-commutative [<=] 68.63	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + \left(-1\right)\right)\right)} \]
      metadata-eval [=>] 68.63	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \color{blue}{-1}\right)\right)} \]

   3. Taylor expanded in t around 0 30.75

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

   4. Simplified 24.96

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

      [Start] 30.75	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 30.88	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 30.88	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 30.88	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 30.88	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 30.88	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 24.96	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 6.99

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

   6. Taylor expanded in k around 0 7.2

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

   7. Simplified 1.23

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

      [Start] 7.2	\[ \frac{2}{\frac{\frac{k \cdot t}{\ell}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}} \]
      associate-*l/ [<=] 1.23	\[ \frac{2}{\frac{\color{blue}{\frac{k}{\ell} \cdot t}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.22

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \frac{\frac{k}{\cos k}}{\ell \cdot {\sin k}^{-2}}\right)}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.84%
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2.5 \cdot 10^{-15} \lor \neg \left(k \leq 6.8 \cdot 10^{-9}\right):\\ \;\;\;\;\ell \cdot \left(\frac{2}{{\sin k}^{2} \cdot \frac{t}{\ell}} \cdot \frac{\cos k}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	15.17%
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{-71} \lor \neg \left(k \leq 8.2 \cdot 10^{-47}\right):\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{\frac{{\sin k}^{2} \cdot \left(k \cdot t\right)}{\frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	5.75%
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{-72} \lor \neg \left(k \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\ell \cdot \frac{2}{k \cdot t}\right) \cdot \left(\ell \cdot \left({\sin k}^{-2} \cdot \frac{\cos k}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 4
Error	1.24%
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2 \cdot 10^{-9} \lor \neg \left(k \leq 10^{-25}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \frac{\frac{k}{\cos k}}{\ell \cdot {\sin k}^{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	20.3%
Cost	20488

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.95 \cdot 10^{+158}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}}\\ \mathbf{elif}\;k \leq -5.5 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq -4.6 \cdot 10^{-8} \lor \neg \left(k \leq 4.8 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 6
Error	12.6%
Cost	20488

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := {\sin k}^{2}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.1 \cdot 10^{-8}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot t}\right)\right) \cdot \frac{t_1}{t_2}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{\frac{t_2 \cdot \left(k \cdot t\right)}{t_1}}\\ \end{array} \]

Alternative 7
Error	6.67%
Cost	20488

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{k \cdot t}\right) \cdot \left(\ell \cdot \left({\sin k}^{-2} \cdot t_1\right)\right)\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\frac{2}{k}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 8
Error	22.18%
Cost	14668

\[\begin{array}{l} t_1 := \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-244}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+194}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\ell}{k} \cdot \left(t \cdot \frac{t}{k}\right)\right)\\ \end{array} \]

Alternative 9
Error	22.12%
Cost	14668

\[\begin{array}{l} t_1 := \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{t}{k} \cdot \left(\ell \cdot \frac{t}{k}\right)\right)\\ \end{array} \]

Alternative 10
Error	20.37%
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2.9 \cdot 10^{-12} \lor \neg \left(k \leq 3.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 11
Error	31.37%
Cost	8009

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -7.5 \cdot 10^{-13} \lor \neg \left(k \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\ell}}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 12
Error	30.65%
Cost	8009

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4.8 \cdot 10^{-10} \lor \neg \left(k \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 13
Error	32.69%
Cost	8004

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \left(\frac{\frac{2}{k}}{\frac{t}{\ell}} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 14
Error	40.19%
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 15
Error	40.17%
Cost	960

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

Alternative 16
Error	39.27%
Cost	960

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

Alternative 17
Error	35.05%
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 2023103 
(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))))