Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 0.7

Time: 27.4s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.02 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{\tan k}{\frac{\ell}{k}} \cdot \sin k\right)}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{t_1}{\frac{\ell}{k \cdot \sin k}}}\\ \end{array} \]

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 (* t (/ k l))))
   (if (<= k -1.02e-11)
     (/ 2.0 (* t_1 (* (/ (tan k) (/ l k)) (sin k))))
     (if (<= k 1.2e-37)
       (* 2.0 (/ (/ (/ l k) k) (* t (* k (/ k l)))))
       (/ 2.0 (* (tan k) (/ t_1 (/ l (* k (sin 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 = t * (k / l);
	double tmp;
	if (k <= -1.02e-11) {
		tmp = 2.0 / (t_1 * ((tan(k) / (l / k)) * sin(k)));
	} else if (k <= 1.2e-37) {
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))));
	} else {
		tmp = 2.0 / (tan(k) * (t_1 / (l / (k * sin(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 = t * (k / l)
    if (k <= (-1.02d-11)) then
        tmp = 2.0d0 / (t_1 * ((tan(k) / (l / k)) * sin(k)))
    else if (k <= 1.2d-37) then
        tmp = 2.0d0 * (((l / k) / k) / (t * (k * (k / l))))
    else
        tmp = 2.0d0 / (tan(k) * (t_1 / (l / (k * sin(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 = t * (k / l);
	double tmp;
	if (k <= -1.02e-11) {
		tmp = 2.0 / (t_1 * ((Math.tan(k) / (l / k)) * Math.sin(k)));
	} else if (k <= 1.2e-37) {
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))));
	} else {
		tmp = 2.0 / (Math.tan(k) * (t_1 / (l / (k * Math.sin(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 = t * (k / l)
	tmp = 0
	if k <= -1.02e-11:
		tmp = 2.0 / (t_1 * ((math.tan(k) / (l / k)) * math.sin(k)))
	elif k <= 1.2e-37:
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))))
	else:
		tmp = 2.0 / (math.tan(k) * (t_1 / (l / (k * math.sin(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(t * Float64(k / l))
	tmp = 0.0
	if (k <= -1.02e-11)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(tan(k) / Float64(l / k)) * sin(k))));
	elseif (k <= 1.2e-37)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * Float64(k * Float64(k / l)))));
	else
		tmp = Float64(2.0 / Float64(tan(k) * Float64(t_1 / Float64(l / Float64(k * sin(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 = t * (k / l);
	tmp = 0.0;
	if (k <= -1.02e-11)
		tmp = 2.0 / (t_1 * ((tan(k) / (l / k)) * sin(k)));
	elseif (k <= 1.2e-37)
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))));
	else
		tmp = 2.0 / (tan(k) * (t_1 / (l / (k * sin(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[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.02e-11], N[(2.0 / N[(t$95$1 * N[(N[(N[Tan[k], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e-37], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t$95$1 / N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.01999999999999994e-11

   1. Initial program 43.1

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

   2. Simplified 35.4

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 43.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 43.1	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 43.1	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 43.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 35.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 35.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 18.7

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

   4. Simplified 13.1

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

      [Start] 18.7	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 18.7	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 18.7	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 13.1	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 13.1	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 13.1	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 4.3

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

   6. Applied egg-rr 39.7

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\tan k \cdot \left(t \cdot k\right)}{\ell} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)} - 1}} \]

   7. Simplified 0.8

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

      [Start] 39.7	\[ \frac{2}{e^{\mathsf{log1p}\left(\frac{\tan k \cdot \left(t \cdot k\right)}{\ell} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)} - 1} \]
      expm1-def [=>] 29.3	\[ \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan k \cdot \left(t \cdot k\right)}{\ell} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)\right)}} \]
      expm1-log1p [=>] 4.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot \left(t \cdot k\right)}{\ell} \cdot \frac{k}{\frac{\ell}{\sin k}}}} \]
      associate-*r/ [=>] 6.2	\[ \frac{2}{\color{blue}{\frac{\frac{\tan k \cdot \left(t \cdot k\right)}{\ell} \cdot k}{\frac{\ell}{\sin k}}}} \]
      associate-/l* [=>] 4.9	\[ \frac{2}{\color{blue}{\frac{\frac{\tan k \cdot \left(t \cdot k\right)}{\ell}}{\frac{\frac{\ell}{\sin k}}{k}}}} \]
      associate-/r* [<=] 4.9	\[ \frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot k\right)}{\ell}}{\color{blue}{\frac{\ell}{\sin k \cdot k}}}} \]
      associate-/l/ [<=] 4.9	\[ \frac{2}{\frac{\frac{\tan k \cdot \left(t \cdot k\right)}{\ell}}{\color{blue}{\frac{\frac{\ell}{k}}{\sin k}}}} \]
      associate-/l/ [=>] 9.1	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot \left(t \cdot k\right)}{\frac{\frac{\ell}{k}}{\sin k} \cdot \ell}}} \]
      times-frac [=>] 4.9	\[ \frac{2}{\color{blue}{\frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}} \cdot \frac{t \cdot k}{\ell}}} \]
      associate-*l/ [<=] 4.3	\[ \frac{2}{\frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}} \]
      *-commutative [=>] 4.3	\[ \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot k\right) \cdot \frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}}}} \]
      *-commutative [=>] 4.3	\[ \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}}} \]
      associate-*r/ [=>] 4.9	\[ \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}}} \]
      associate-*l/ [<=] 0.8	\[ \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot t\right)} \cdot \frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}}} \]
      *-commutative [=>] 0.8	\[ \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\ell}\right)} \cdot \frac{\tan k}{\frac{\frac{\ell}{k}}{\sin k}}} \]
      associate-/r/ [=>] 0.8	\[ \frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot \sin k\right)}} \]

   if -1.01999999999999994e-11 < k < 1.19999999999999995e-37

   1. Initial program 62.7

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

   2. Simplified 52.0

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

      [Start] 62.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 62.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 62.6	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 62.6	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 61.4	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 61.4	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 61.4	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 61.9	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 61.9	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 61.9	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 47.6

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

   4. Simplified 44.2

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

      [Start] 47.6	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 47.6	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 44.2	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 19.7

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

   6. Applied egg-rr 0.5

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

   if 1.19999999999999995e-37 < k

   1. Initial program 45.4

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

   2. Simplified 37.0

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 45.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 45.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 45.4	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 45.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 37.0	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 37.0	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 19.0

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

   4. Simplified 12.6

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

      [Start] 19.0	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 19.0	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.0	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 12.6	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 12.6	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 12.6	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 4.2

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

   6. Taylor expanded in t around 0 4.6

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

   7. Simplified 0.8

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

      [Start] 4.6	\[ \frac{2}{\tan k \cdot \frac{\frac{k \cdot t}{\ell}}{\frac{\ell}{k \cdot \sin k}}} \]
      associate-*l/ [<=] 0.8	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\frac{k}{\ell} \cdot t}}{\frac{\ell}{k \cdot \sin k}}} \]
      *-commutative [=>] 0.8	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{t \cdot \frac{k}{\ell}}}{\frac{\ell}{k \cdot \sin k}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.02 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{\tan k}{\frac{\ell}{k}} \cdot \sin k\right)}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{t \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot \sin k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.8
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+51}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{t \cdot \sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	10.8
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-9} \lor \neg \left(k \leq 4 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3
Error	5.2
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -6.5 \cdot 10^{-8} \lor \neg \left(k \leq 10^{-37}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot \sin k}{\frac{\ell}{k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.75 \cdot 10^{-9} \lor \neg \left(k \leq 1.2 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{t \cdot \frac{k}{\ell}}{\frac{\ell}{k \cdot \sin k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 5
Error	24.0
Cost	1224

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-188}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{elif}\;\ell \leq 10^{-179}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 6
Error	22.3
Cost	1088

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ 2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Alternative 7
Error	25.4
Cost	960

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

Alternative 8
Error	25.4
Cost	960

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

Alternative 9
Error	24.0
Cost	960

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

Alternative 10
Error	24.0
Cost	960

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

Alternative 11
Error	22.2
Cost	960

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

Reproduce

herbie shell --seed 2023034 
(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))))