Toniolo and Linder, Equation (10-)

Average Error: 47.3 → 4.4

Time: 39.3s

Precision: binary64

Cost: 20617

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{+214} \lor \neg \left(k \leq 1.6 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{\frac{\cos k}{k}}{-t} \cdot \frac{\frac{\ell}{-k}}{\frac{{\sin k}^{2}}{\ell \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{2}{\sin k}\right) \cdot \frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t}\\ \end{array} \]

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 (<= k -8.5e+214) (not (<= k 1.6e+125)))
   (* (/ (/ (cos k) k) (- t)) (/ (/ l (- k)) (/ (pow (sin k) 2.0) (* l 2.0))))
   (* (* (/ l (sin k)) (/ 2.0 (sin k))) (/ (/ (cos k) (/ k (/ l k))) t))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

↓

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -8.5e+214) || !(k <= 1.6e+125)) {
		tmp = ((cos(k) / k) / -t) * ((l / -k) / (pow(sin(k), 2.0) / (l * 2.0)));
	} else {
		tmp = ((l / sin(k)) * (2.0 / sin(k))) * ((cos(k) / (k / (l / k))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= (-8.5d+214)) .or. (.not. (k <= 1.6d+125))) then
        tmp = ((cos(k) / k) / -t) * ((l / -k) / ((sin(k) ** 2.0d0) / (l * 2.0d0)))
    else
        tmp = ((l / sin(k)) * (2.0d0 / sin(k))) * ((cos(k) / (k / (l / k))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

↓

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -8.5e+214) || !(k <= 1.6e+125)) {
		tmp = ((Math.cos(k) / k) / -t) * ((l / -k) / (Math.pow(Math.sin(k), 2.0) / (l * 2.0)));
	} else {
		tmp = ((l / Math.sin(k)) * (2.0 / Math.sin(k))) * ((Math.cos(k) / (k / (l / k))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

↓

def code(t, l, k):
	tmp = 0
	if (k <= -8.5e+214) or not (k <= 1.6e+125):
		tmp = ((math.cos(k) / k) / -t) * ((l / -k) / (math.pow(math.sin(k), 2.0) / (l * 2.0)))
	else:
		tmp = ((l / math.sin(k)) * (2.0 / math.sin(k))) * ((math.cos(k) / (k / (l / k))) / t)
	return tmp

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

↓

function code(t, l, k)
	tmp = 0.0
	if ((k <= -8.5e+214) || !(k <= 1.6e+125))
		tmp = Float64(Float64(Float64(cos(k) / k) / Float64(-t)) * Float64(Float64(l / Float64(-k)) / Float64((sin(k) ^ 2.0) / Float64(l * 2.0))));
	else
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(2.0 / sin(k))) * Float64(Float64(cos(k) / Float64(k / Float64(l / k))) / t));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

↓

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -8.5e+214) || ~((k <= 1.6e+125)))
		tmp = ((cos(k) / k) / -t) * ((l / -k) / ((sin(k) ^ 2.0) / (l * 2.0)));
	else
		tmp = ((l / sin(k)) * (2.0 / sin(k))) * ((cos(k) / (k / (l / k))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := If[Or[LessEqual[k, -8.5e+214], N[Not[LessEqual[k, 1.6e+125]], $MachinePrecision]], N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / (-t)), $MachinePrecision] * N[(N[(l / (-k)), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -8.50000000000000045e214 or 1.59999999999999992e125 < k

   1. Initial program 38.7

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

   2. Simplified 31.3

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

      [Start] 38.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 38.7	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 38.7	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 38.7	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 36.9	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 36.9	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 36.9	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 31.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 31.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 31.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around inf 22.5

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

   4. Simplified 20.9

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

      [Start] 22.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 22.9	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      associate-*r* [=>] 22.9	\[ \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2}}\right) \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 22.9	\[ \left(2 \cdot \frac{\cos k}{\color{blue}{k \cdot k}}\right) \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 22.9	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t} \]
      *-commutative [=>] 22.9	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}} \]
      times-frac [=>] 20.9	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)} \]

   5. Applied egg-rr 19.8

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

   6. Taylor expanded in k around inf 22.5

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

   7. Simplified 12.6

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

      [Start] 22.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 22.9	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 22.9	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      associate-*r* [=>] 22.9	\[ \color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 22.9	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t} \]
      associate-*r/ [<=] 20.9	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{\sin k}^{2} \cdot t}\right)} \]
      associate-/r* [=>] 20.9	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\ell}{{\sin k}^{2}}}{t}}\right) \]
      associate-*r* [=>] 19.9	\[ \color{blue}{\left(\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell\right) \cdot \frac{\frac{\ell}{{\sin k}^{2}}}{t}} \]
      associate-*r/ [=>] 19.6	\[ \color{blue}{\frac{\left(\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell\right) \cdot \frac{\ell}{{\sin k}^{2}}}{t}} \]
      associate-*l/ [<=] 19.5	\[ \color{blue}{\frac{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}} \]
      *-commutative [=>] 19.5	\[ \color{blue}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell}{t}} \]
      associate-*r/ [<=] 19.7	\[ \frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\left(\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell}{t}\right)} \]
      associate-*l* [=>] 19.7	\[ \frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\left(2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)} \]
      associate-*r* [=>] 19.7	\[ \color{blue}{\left(\frac{\ell}{{\sin k}^{2}} \cdot 2\right) \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]

   8. Applied egg-rr 12.8

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

   9. Simplified 6.6

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

      [Start] 12.8	\[ \frac{\frac{\cos k}{k} \cdot \frac{\ell}{-k}}{\left(-t\right) \cdot \frac{{\sin k}^{2}}{2 \cdot \ell}} \]
      times-frac [=>] 6.6	\[ \color{blue}{\frac{\frac{\cos k}{k}}{-t} \cdot \frac{\frac{\ell}{-k}}{\frac{{\sin k}^{2}}{2 \cdot \ell}}} \]
      *-commutative [=>] 6.6	\[ \frac{\frac{\cos k}{k}}{-t} \cdot \frac{\frac{\ell}{-k}}{\frac{{\sin k}^{2}}{\color{blue}{\ell \cdot 2}}} \]

   if -8.50000000000000045e214 < k < 1.59999999999999992e125

   1. Initial program 52.6

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

   2. Simplified 40.5

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

      [Start] 52.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 52.6	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 52.6	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 52.7	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 51.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 51.5	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 51.5	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 40.5	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 40.5	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 40.5	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around inf 21.8

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

   4. Simplified 13.5

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

      [Start] 21.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 21.2	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      associate-*r* [=>] 21.2	\[ \color{blue}{\left(2 \cdot \frac{\cos k}{{k}^{2}}\right) \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 21.2	\[ \left(2 \cdot \frac{\cos k}{\color{blue}{k \cdot k}}\right) \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 21.2	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t} \]
      *-commutative [=>] 21.2	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}} \]
      times-frac [=>] 13.5	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)} \]

   5. Applied egg-rr 7.6

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

   6. Taylor expanded in k around inf 21.8

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

   7. Simplified 5.5

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

      [Start] 21.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 21.2	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 21.2	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      associate-*r* [=>] 21.2	\[ \color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 21.2	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t} \]
      associate-*r/ [<=] 14.7	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{\sin k}^{2} \cdot t}\right)} \]
      associate-/r* [=>] 12.8	\[ \left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\ell}{{\sin k}^{2}}}{t}}\right) \]
      associate-*r* [=>] 7.7	\[ \color{blue}{\left(\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell\right) \cdot \frac{\frac{\ell}{{\sin k}^{2}}}{t}} \]
      associate-*r/ [=>] 13.6	\[ \color{blue}{\frac{\left(\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell\right) \cdot \frac{\ell}{{\sin k}^{2}}}{t}} \]
      associate-*l/ [<=] 7.5	\[ \color{blue}{\frac{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}} \]
      *-commutative [=>] 7.5	\[ \color{blue}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \ell}{t}} \]
      associate-*r/ [<=] 9.8	\[ \frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\left(\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{\ell}{t}\right)} \]
      associate-*l* [=>] 9.8	\[ \frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\left(2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)} \]
      associate-*r* [=>] 9.8	\[ \color{blue}{\left(\frac{\ell}{{\sin k}^{2}} \cdot 2\right) \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t}\right)} \]

   8. Applied egg-rr 3.0

      \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{2}{\sin k}\right)} \cdot \frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{+214} \lor \neg \left(k \leq 1.6 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{\frac{\cos k}{k}}{-t} \cdot \frac{\frac{\ell}{-k}}{\frac{{\sin k}^{2}}{\ell \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{2}{\sin k}\right) \cdot \frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	20753

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-83} \lor \neg \left(\ell \leq 4 \cdot 10^{-164}\right) \land \ell \leq 4.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\cos k \cdot \frac{\ell}{k \cdot k}}{t}\\ \end{array} \]

Alternative 2
Error	3.7
Cost	20489

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+142} \lor \neg \left(t \leq 10^{-15}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot 2}{k}}{t \cdot \frac{t_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \end{array} \]

Alternative 3
Error	6.6
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{t_1}\right) \cdot \frac{\cos k \cdot \frac{\ell}{k \cdot k}}{t}\\ \end{array} \]

Alternative 4
Error	6.4
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell \cdot 2}{{\sin k}^{2}}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;t_1 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;t \leq 3.95 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t} \cdot t_1\\ \end{array} \]

Alternative 5
Error	6.4
Cost	20488

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_2} \cdot \left(t_1 \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_2 \cdot \left(t \cdot \frac{\frac{k}{\ell}}{t_1}\right)}\\ \end{array} \]

Alternative 6
Error	6.4
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{\frac{t \cdot t_1}{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}}\\ \end{array} \]

Alternative 7
Error	6.4
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\cos k}{k \cdot k}\right)}{t \cdot \frac{t_1}{\ell}}\\ \end{array} \]

Alternative 8
Error	3.4
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -120000:\\ \;\;\;\;\frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t \cdot \frac{t_1}{\ell \cdot 2}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot 2}{k}}{t \cdot \frac{t_1}{\ell}}\\ \end{array} \]

Alternative 9
Error	3.4
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -135000:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{2}{\sin k}\right) \cdot \frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{elif}\;t \leq 10^{-15}:\\ \;\;\;\;\frac{\ell \cdot 2}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot 2}{k}}{t \cdot \frac{t_1}{\ell}}\\ \end{array} \]

Alternative 10
Error	13.9
Cost	14800

\[\begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{t}{k} \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\\ t_2 := \frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\frac{\ell}{k}}}}{t \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	13.8
Cost	14800

\[\begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}\\ t_2 := \frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+161}:\\ \;\;\;\;t_1 \cdot \left(\frac{t}{k} \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\frac{\ell}{k}}}}{t \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}} \cdot \left(t \cdot \frac{t}{k}\right)\\ \end{array} \]

Alternative 12
Error	13.6
Cost	14800

\[\begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}\\ t_2 := \frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{t_1}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\frac{\ell}{k}}}}{t \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}} \cdot \left(t \cdot \frac{t}{k}\right)\\ \end{array} \]

Alternative 13
Error	14.1
Cost	14792

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t} \cdot \mathsf{fma}\left(2, \frac{\ell}{k \cdot k}, \ell \cdot 0.6666666666666666\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+291}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)} \cdot \left(t \cdot \frac{\ell}{k \cdot \frac{k}{t}}\right)\\ \end{array} \]

Alternative 14
Error	14.4
Cost	14537

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3 \cdot 10^{-302} \lor \neg \left(\ell \cdot \ell \leq 10^{+291}\right):\\ \;\;\;\;\frac{\frac{\cos k}{\frac{k}{\frac{\ell}{k}}}}{t} \cdot \mathsf{fma}\left(2, \frac{\ell}{k \cdot k}, \ell \cdot 0.6666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \end{array} \]

Alternative 15
Error	15.9
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.05 \cdot 10^{-5} \lor \neg \left(k \leq 1.05 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\frac{\ell}{k}}}}{t \cdot \frac{{\sin k}^{2}}{\ell}}\\ \end{array} \]

Alternative 16
Error	23.4
Cost	13696

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

Alternative 17
Error	23.2
Cost	13696

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

Alternative 18
Error	24.2
Cost	7488

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

Alternative 19
Error	30.3
Cost	960

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

Alternative 20
Error	29.7
Cost	960

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

Alternative 21
Error	26.1
Cost	960

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

Reproduce

herbie shell --seed 2023011 
(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))))