Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 5.2

Time: 25.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\cos k}{t}\\ t_2 := \ell \cdot t_1\\ t_3 := \frac{t_1 \cdot 2}{\sin k}\\ t_4 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -4.4355977450628285 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\sin k} \cdot t_3\\ \mathbf{elif}\;k \leq -9.950957702371115 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{t_4}\right)\\ \mathbf{elif}\;k \leq 1.0932265067334292 \cdot 10^{-92}:\\ \;\;\;\;t_3 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{elif}\;k \leq 7.988200189249557 \cdot 10^{+105}:\\ \;\;\;\;\frac{2 \cdot \frac{t_2}{\frac{k \cdot k}{\ell}}}{t_4}\\ \mathbf{elif}\;k \leq 2.0585063427582685 \cdot 10^{+208}:\\ \;\;\;\;\frac{2 \cdot \frac{1}{\frac{t}{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{k}}{t_4}\\ \end{array} \]

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 (/ (cos k) t))
        (t_2 (* l t_1))
        (t_3 (/ (* t_1 2.0) (sin k)))
        (t_4 (pow (sin k) 2.0)))
   (if (<= k -4.4355977450628285e+138)
     (* (/ (/ 1.0 (* (/ k l) (/ k l))) (sin k)) t_3)
     (if (<= k -9.950957702371115e-5)
       (* 2.0 (* (* (/ l (* k k)) (/ l t)) (/ (cos k) t_4)))
       (if (<= k 1.0932265067334292e-92)
         (* t_3 (* (/ l k) (/ (/ l k) (sin k))))
         (if (<= k 7.988200189249557e+105)
           (/ (* 2.0 (/ t_2 (/ (* k k) l))) t_4)
           (if (<= k 2.0585063427582685e+208)
             (/ (* 2.0 (/ 1.0 (/ t (* (cos k) (pow (/ l k) 2.0))))) t_4)
             (/ (* 2.0 (/ (* (/ l k) t_2) k)) t_4))))))))

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 = cos(k) / t;
	double t_2 = l * t_1;
	double t_3 = (t_1 * 2.0) / sin(k);
	double t_4 = pow(sin(k), 2.0);
	double tmp;
	if (k <= -4.4355977450628285e+138) {
		tmp = ((1.0 / ((k / l) * (k / l))) / sin(k)) * t_3;
	} else if (k <= -9.950957702371115e-5) {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (cos(k) / t_4));
	} else if (k <= 1.0932265067334292e-92) {
		tmp = t_3 * ((l / k) * ((l / k) / sin(k)));
	} else if (k <= 7.988200189249557e+105) {
		tmp = (2.0 * (t_2 / ((k * k) / l))) / t_4;
	} else if (k <= 2.0585063427582685e+208) {
		tmp = (2.0 * (1.0 / (t / (cos(k) * pow((l / k), 2.0))))) / t_4;
	} else {
		tmp = (2.0 * (((l / k) * t_2) / k)) / t_4;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos(k) / t
    t_2 = l * t_1
    t_3 = (t_1 * 2.0d0) / sin(k)
    t_4 = sin(k) ** 2.0d0
    if (k <= (-4.4355977450628285d+138)) then
        tmp = ((1.0d0 / ((k / l) * (k / l))) / sin(k)) * t_3
    else if (k <= (-9.950957702371115d-5)) then
        tmp = 2.0d0 * (((l / (k * k)) * (l / t)) * (cos(k) / t_4))
    else if (k <= 1.0932265067334292d-92) then
        tmp = t_3 * ((l / k) * ((l / k) / sin(k)))
    else if (k <= 7.988200189249557d+105) then
        tmp = (2.0d0 * (t_2 / ((k * k) / l))) / t_4
    else if (k <= 2.0585063427582685d+208) then
        tmp = (2.0d0 * (1.0d0 / (t / (cos(k) * ((l / k) ** 2.0d0))))) / t_4
    else
        tmp = (2.0d0 * (((l / k) * t_2) / k)) / t_4
    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 = Math.cos(k) / t;
	double t_2 = l * t_1;
	double t_3 = (t_1 * 2.0) / Math.sin(k);
	double t_4 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= -4.4355977450628285e+138) {
		tmp = ((1.0 / ((k / l) * (k / l))) / Math.sin(k)) * t_3;
	} else if (k <= -9.950957702371115e-5) {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (Math.cos(k) / t_4));
	} else if (k <= 1.0932265067334292e-92) {
		tmp = t_3 * ((l / k) * ((l / k) / Math.sin(k)));
	} else if (k <= 7.988200189249557e+105) {
		tmp = (2.0 * (t_2 / ((k * k) / l))) / t_4;
	} else if (k <= 2.0585063427582685e+208) {
		tmp = (2.0 * (1.0 / (t / (Math.cos(k) * Math.pow((l / k), 2.0))))) / t_4;
	} else {
		tmp = (2.0 * (((l / k) * t_2) / k)) / t_4;
	}
	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 = math.cos(k) / t
	t_2 = l * t_1
	t_3 = (t_1 * 2.0) / math.sin(k)
	t_4 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= -4.4355977450628285e+138:
		tmp = ((1.0 / ((k / l) * (k / l))) / math.sin(k)) * t_3
	elif k <= -9.950957702371115e-5:
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (math.cos(k) / t_4))
	elif k <= 1.0932265067334292e-92:
		tmp = t_3 * ((l / k) * ((l / k) / math.sin(k)))
	elif k <= 7.988200189249557e+105:
		tmp = (2.0 * (t_2 / ((k * k) / l))) / t_4
	elif k <= 2.0585063427582685e+208:
		tmp = (2.0 * (1.0 / (t / (math.cos(k) * math.pow((l / k), 2.0))))) / t_4
	else:
		tmp = (2.0 * (((l / k) * t_2) / k)) / t_4
	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(cos(k) / t)
	t_2 = Float64(l * t_1)
	t_3 = Float64(Float64(t_1 * 2.0) / sin(k))
	t_4 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= -4.4355977450628285e+138)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(k / l) * Float64(k / l))) / sin(k)) * t_3);
	elseif (k <= -9.950957702371115e-5)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * Float64(cos(k) / t_4)));
	elseif (k <= 1.0932265067334292e-92)
		tmp = Float64(t_3 * Float64(Float64(l / k) * Float64(Float64(l / k) / sin(k))));
	elseif (k <= 7.988200189249557e+105)
		tmp = Float64(Float64(2.0 * Float64(t_2 / Float64(Float64(k * k) / l))) / t_4);
	elseif (k <= 2.0585063427582685e+208)
		tmp = Float64(Float64(2.0 * Float64(1.0 / Float64(t / Float64(cos(k) * (Float64(l / k) ^ 2.0))))) / t_4);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) * t_2) / k)) / t_4);
	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 = cos(k) / t;
	t_2 = l * t_1;
	t_3 = (t_1 * 2.0) / sin(k);
	t_4 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= -4.4355977450628285e+138)
		tmp = ((1.0 / ((k / l) * (k / l))) / sin(k)) * t_3;
	elseif (k <= -9.950957702371115e-5)
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (cos(k) / t_4));
	elseif (k <= 1.0932265067334292e-92)
		tmp = t_3 * ((l / k) * ((l / k) / sin(k)));
	elseif (k <= 7.988200189249557e+105)
		tmp = (2.0 * (t_2 / ((k * k) / l))) / t_4;
	elseif (k <= 2.0585063427582685e+208)
		tmp = (2.0 * (1.0 / (t / (cos(k) * ((l / k) ^ 2.0))))) / t_4;
	else
		tmp = (2.0 * (((l / k) * t_2) / k)) / t_4;
	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[Cos[k], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(l * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, -4.4355977450628285e+138], N[(N[(N[(1.0 / N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[k, -9.950957702371115e-5], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0932265067334292e-92], N[(t$95$3 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.988200189249557e+105], N[(N[(2.0 * N[(t$95$2 / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[k, 2.0585063427582685e+208], N[(N[(2.0 * N[(1.0 / N[(t / N[(N[Cos[k], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * t$95$2), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 6 regimes

2. if k < -4.4355977450628285e138

   1. Initial program 39.4

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

   2. Simplified 33.5

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

   3. Taylor expanded in t around 0 23.0

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

   4. Simplified 18.5

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

   5. Taylor expanded in k around inf 23.0

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

   6. Simplified 20.0

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

   7. Applied egg-rr 3.5

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

   8. Applied egg-rr 3.6

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}}{\sin k} \cdot \frac{\frac{\cos k}{t} \cdot 2}{\sin k} \]

   if -4.4355977450628285e138 < k < -9.95095770237111463e-5

   1. Initial program 51.0

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

   2. Simplified 39.6

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

   3. Taylor expanded in t around 0 13.3

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

   4. Simplified 13.3

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

   5. Taylor expanded in k around inf 13.3

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

   6. Simplified 8.7

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

   7. Applied egg-rr 8.8

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

   8. Taylor expanded in l around 0 13.3

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

   9. Simplified 2.4

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

   if -9.95095770237111463e-5 < k < 1.0932265067334292e-92

   1. Initial program 62.4

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

   2. Simplified 58.4

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

   3. Taylor expanded in t around 0 40.4

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

   4. Simplified 40.8

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

   5. Taylor expanded in k around inf 40.4

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

   6. Simplified 25.0

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

   7. Applied egg-rr 12.4

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

   8. Applied egg-rr 9.9

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

   if 1.0932265067334292e-92 < k < 7.98820018924955744e105

   1. Initial program 53.9

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

   2. Simplified 42.2

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

   3. Taylor expanded in t around 0 16.6

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

   4. Simplified 16.7

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

   5. Taylor expanded in k around inf 16.6

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

   6. Simplified 12.1

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

   7. Applied egg-rr 3.6

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

   if 7.98820018924955744e105 < k < 2.05850634275826847e208

   1. Initial program 44.9

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

   2. Simplified 34.7

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

   3. Taylor expanded in t around 0 20.9

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

   4. Simplified 15.8

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

   5. Taylor expanded in k around inf 20.9

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

   6. Simplified 15.5

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

   7. Applied egg-rr 4.9

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

   if 2.05850634275826847e208 < k

   1. Initial program 35.4

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

   2. Simplified 32.1

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

   3. Taylor expanded in t around 0 22.8

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

   4. Simplified 19.6

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

   5. Taylor expanded in k around inf 22.8

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

   6. Simplified 20.5

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

   7. Applied egg-rr 7.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\cos k}{t}\right)}{k}} \cdot 2}{{\sin k}^{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.4355977450628285 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{\sin k} \cdot \frac{\frac{\cos k}{t} \cdot 2}{\sin k}\\ \mathbf{elif}\;k \leq -9.950957702371115 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq 1.0932265067334292 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\cos k}{t} \cdot 2}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{elif}\;k \leq 7.988200189249557 \cdot 10^{+105}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell \cdot \frac{\cos k}{t}}{\frac{k \cdot k}{\ell}}}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq 2.0585063427582685 \cdot 10^{+208}:\\ \;\;\;\;\frac{2 \cdot \frac{1}{\frac{t}{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\cos k}{t}\right)}{k}}{{\sin k}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))