Toniolo and Linder, Equation (10+)

Average Error: 32.1 → 17.0

Time: 39.8s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\left({t}^{3} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \frac{{\left(\frac{k}{t}\right)}^{2}}{2}\right)\right)}\\ \mathbf{if}\;t \leq -1.22 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 130:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          l
          (*
           (* (pow t 3.0) (/ (sin k) l))
           (* (tan k) (+ 1.0 (/ (pow (/ k t) 2.0) 2.0)))))))
   (if (<= t -1.22e-67)
     t_1
     (if (<= t 130.0)
       (* (/ 2.0 (pow (sin k) 2.0)) (/ l (* (/ t l) (/ (pow k 2.0) (cos k)))))
       t_1))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = l / ((pow(t, 3.0) * (sin(k) / l)) * (tan(k) * (1.0 + (pow((k / t), 2.0) / 2.0))));
	double tmp;
	if (t <= -1.22e-67) {
		tmp = t_1;
	} else if (t <= 130.0) {
		tmp = (2.0 / pow(sin(k), 2.0)) * (l / ((t / l) * (pow(k, 2.0) / cos(k))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (((t ** 3.0d0) * (sin(k) / l)) * (tan(k) * (1.0d0 + (((k / t) ** 2.0d0) / 2.0d0))))
    if (t <= (-1.22d-67)) then
        tmp = t_1
    else if (t <= 130.0d0) then
        tmp = (2.0d0 / (sin(k) ** 2.0d0)) * (l / ((t / l) * ((k ** 2.0d0) / cos(k))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

↓

public static double code(double t, double l, double k) {
	double t_1 = l / ((Math.pow(t, 3.0) * (Math.sin(k) / l)) * (Math.tan(k) * (1.0 + (Math.pow((k / t), 2.0) / 2.0))));
	double tmp;
	if (t <= -1.22e-67) {
		tmp = t_1;
	} else if (t <= 130.0) {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (l / ((t / l) * (Math.pow(k, 2.0) / Math.cos(k))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(t, l, k):
	t_1 = l / ((math.pow(t, 3.0) * (math.sin(k) / l)) * (math.tan(k) * (1.0 + (math.pow((k / t), 2.0) / 2.0))))
	tmp = 0
	if t <= -1.22e-67:
		tmp = t_1
	elif t <= 130.0:
		tmp = (2.0 / math.pow(math.sin(k), 2.0)) * (l / ((t / l) * (math.pow(k, 2.0) / math.cos(k))))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(t, l, k)
	t_1 = Float64(l / Float64(Float64((t ^ 3.0) * Float64(sin(k) / l)) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t) ^ 2.0) / 2.0)))))
	tmp = 0.0
	if (t <= -1.22e-67)
		tmp = t_1;
	elseif (t <= 130.0)
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(l / Float64(Float64(t / l) * Float64((k ^ 2.0) / cos(k)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(t, l, k)
	t_1 = l / (((t ^ 3.0) * (sin(k) / l)) * (tan(k) * (1.0 + (((k / t) ^ 2.0) / 2.0))));
	tmp = 0.0;
	if (t <= -1.22e-67)
		tmp = t_1;
	elseif (t <= 130.0)
		tmp = (2.0 / (sin(k) ^ 2.0)) * (l / ((t / l) * ((k ^ 2.0) / cos(k))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.22e-67], t$95$1, If[LessEqual[t, 130.0], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.22e-67 or 130 < t

   1. Initial program 22.4

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

   2. Simplified 21.3

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

      [Start] 22.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-28 [=>] 22.3	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-15 [=>] 22.3	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      rational.json-simplify-50 [=>] 22.3	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]
      rational.json-simplify-50 [=>] 22.3	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\tan k \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}} \]
      rational.json-simplify-41 [=>] 22.2	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\tan k \cdot \color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
      rational.json-simplify-41 [=>] 22.2	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\tan k}{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}}} \]
      rational.json-simplify-27 [=>] 22.2	\[ \color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \cdot \frac{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}{\tan k}} \]
      rational.json-simplify-1 [=>] 22.2	\[ \frac{2}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}{\tan k} \]
      rational.json-simplify-1 [=>] 22.2	\[ \frac{2}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \cdot \frac{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}{\tan k} \]
      rational.json-simplify-2 [=>] 22.2	\[ \frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \cdot \frac{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}{\tan k} \]
      metadata-eval [=>] 22.2	\[ \frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \cdot \frac{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}{\tan k} \]
      rational.json-simplify-1 [=>] 22.2	\[ \frac{2}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}{\tan k} \]
      rational.json-simplify-14 [=>] 21.3	\[ \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}}}{\tan k} \]

   3. Applied egg-rr 15.9

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

   if -1.22e-67 < t < 130

   1. Initial program 52.7

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

   2. Simplified 53.1

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

      [Start] 52.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-50 [=>] 52.7	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational.json-simplify-50 [=>] 52.7	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
      rational.json-simplify-24 [=>] 53.2	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational.json-simplify-24 [=>] 53.1	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational.json-simplify-19 [<=] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot 1\right)}\right)} \]
      rational.json-simplify-50 [<=] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(1 \cdot \left(\tan k \cdot \sin k\right)\right)}\right)} \]
      rational.json-simplify-1 [=>] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(1 \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]
      rational.json-simplify-1 [=>] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}\right) \cdot \left(1 \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]
      rational.json-simplify-2 [=>] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)} \cdot \left(1 \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]
      metadata-eval [=>] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right) \cdot \left(1 \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]
      rational.json-simplify-1 [=>] 53.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(1 \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]

   3. Applied egg-rr 51.7

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

   4. Taylor expanded in t around 0 23.4

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

   5. Applied egg-rr 21.6

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}}} \]

   6. Simplified 19.6

      \[\leadsto \frac{2}{\color{blue}{{\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right) + {\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right)}} \]
      Proof

      [Start] 21.6	\[ \frac{2}{\left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-50 [=>] 21.6	\[ \frac{2}{\color{blue}{\frac{0.5}{\ell} \cdot \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right)} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-48 [=>] 21.6	\[ \frac{2}{\frac{0.5}{\ell} \cdot \color{blue}{\frac{\left({\sin k}^{2} \cdot \frac{t}{\ell}\right) \cdot {k}^{2}}{\cos k}} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-50 [<=] 21.6	\[ \frac{2}{\frac{0.5}{\ell} \cdot \frac{\color{blue}{{k}^{2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}}{\cos k} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-48 [<=] 21.6	\[ \frac{2}{\frac{0.5}{\ell} \cdot \color{blue}{\left(\left({\sin k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{{k}^{2}}{\cos k}\right)} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-50 [=>] 21.6	\[ \frac{2}{\frac{0.5}{\ell} \cdot \color{blue}{\left(\frac{{k}^{2}}{\cos k} \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right)} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-24 [=>] 21.5	\[ \frac{2}{\frac{0.5}{\ell} \cdot \color{blue}{\left({\sin k}^{2} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right)} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-24 [=>] 21.3	\[ \frac{2}{\color{blue}{{\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right)} + \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right) \cdot \frac{0.5}{\ell}} \]
      rational.json-simplify-50 [=>] 21.3	\[ \frac{2}{{\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right) + \color{blue}{\frac{0.5}{\ell} \cdot \left({k}^{2} \cdot \frac{{\sin k}^{2} \cdot \frac{t}{\ell}}{\cos k}\right)}} \]
      rational.json-simplify-48 [=>] 21.3	\[ \frac{2}{{\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right) + \frac{0.5}{\ell} \cdot \color{blue}{\frac{\left({\sin k}^{2} \cdot \frac{t}{\ell}\right) \cdot {k}^{2}}{\cos k}}} \]
      rational.json-simplify-50 [<=] 21.3	\[ \frac{2}{{\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right) + \frac{0.5}{\ell} \cdot \frac{\color{blue}{{k}^{2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}}{\cos k}} \]
      rational.json-simplify-48 [<=] 21.3	\[ \frac{2}{{\sin k}^{2} \cdot \left(\frac{0.5}{\ell} \cdot \left(\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right) + \frac{0.5}{\ell} \cdot \color{blue}{\left(\left({\sin k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{{k}^{2}}{\cos k}\right)}} \]

   7. Applied egg-rr 19.7

      \[\leadsto \color{blue}{\frac{\frac{2}{{\sin k}^{2}}}{\frac{\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}}{\ell}} + 0} \]

   8. Simplified 19.4

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

      [Start] 19.7	\[ \frac{\frac{2}{{\sin k}^{2}}}{\frac{\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}}{\ell}} + 0 \]
      rational.json-simplify-8 [=>] 19.7	\[ \color{blue}{\frac{\frac{2}{{\sin k}^{2}}}{\frac{\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}}{\ell}}} \]
      rational.json-simplify-27 [=>] 19.4	\[ \color{blue}{\frac{2}{{\sin k}^{2}} \cdot \frac{\ell}{\frac{{k}^{2}}{\cos k} \cdot \frac{t}{\ell}}} \]
      rational.json-simplify-50 [=>] 19.4	\[ \frac{2}{{\sin k}^{2}} \cdot \frac{\ell}{\color{blue}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{-67}:\\ \;\;\;\;\frac{\ell}{\left({t}^{3} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \frac{{\left(\frac{k}{t}\right)}^{2}}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 130:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left({t}^{3} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \frac{{\left(\frac{k}{t}\right)}^{2}}{2}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.3
Cost	27088

\[\begin{array}{l} t_1 := \frac{2}{{\sin k}^{2}} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{if}\;k \leq -58000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq -5.5 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.9
Cost	27088

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{t_1} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{if}\;k \leq -13600000:\\ \;\;\;\;\frac{2}{\frac{\frac{t_1}{\ell}}{\cos k} \cdot \frac{{k}^{2} \cdot t}{\ell}}\\ \mathbf{elif}\;k \leq -3.55 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq -7.4 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	21.0
Cost	27088

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -560000000:\\ \;\;\;\;\frac{2}{\frac{\frac{t_1}{\ell}}{\cos k} \cdot \frac{{k}^{2} \cdot t}{\ell}}\\ \mathbf{elif}\;k \leq -3.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq -7.4 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\ell} \cdot \frac{t_1 \cdot \frac{t}{\ell}}{\cos k}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 4
Error	20.9
Cost	27088

\[\begin{array}{l} t_1 := \frac{{k}^{2}}{\cos k}\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{2}{t_2}\\ \mathbf{if}\;k \leq -18500000:\\ \;\;\;\;\frac{2}{\frac{\frac{t_2}{\ell}}{\cos k} \cdot \frac{{k}^{2} \cdot t}{\ell}}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq -7.4 \cdot 10^{-112}:\\ \;\;\;\;\frac{t_3}{\frac{t_1 \cdot \frac{t}{\ell}}{\ell}} - 0\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{\ell}{\frac{t}{\ell} \cdot t_1}\\ \end{array} \]

Alternative 5
Error	17.1
Cost	27080

\[\begin{array}{l} t_1 := \frac{2}{\tan k} \cdot \left(\ell \cdot \frac{\frac{\ell}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	22.1
Cost	26696

\[\begin{array}{l} t_1 := \frac{k \cdot {t}^{3}}{\ell}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \left(\tan k \cdot \left(1 + \frac{{\left(\frac{k}{t}\right)}^{2}}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 135:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\sin k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot k}\\ \end{array} \]

Alternative 7
Error	21.7
Cost	26696

\[\begin{array}{l} t_1 := \frac{k \cdot {t}^{3}}{\ell}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \left(\tan k \cdot \left(1 + \frac{{\left(\frac{k}{t}\right)}^{2}}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 260:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{\sin k \cdot \left({k}^{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot k}\\ \end{array} \]

Alternative 8
Error	21.5
Cost	26696

\[\begin{array}{l} t_1 := \frac{k \cdot {t}^{3}}{\ell}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \left(\tan k \cdot \left(1 + \frac{{\left(\frac{k}{t}\right)}^{2}}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 500:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\sin k \cdot \left(\tan k \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot k}\\ \end{array} \]

Alternative 9
Error	24.2
Cost	20680

\[\begin{array}{l} t_1 := \frac{2}{\tan k} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.9
Cost	20680

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := k \cdot {t}^{3}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell}{\frac{t_2}{\ell} \cdot \left(\tan k \cdot \left(1 + \frac{t_1}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell \cdot \frac{\ell}{t_2}}{2 + t_1}\\ \end{array} \]

Alternative 11
Error	23.9
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{\left({t}^{3} \cdot \frac{k}{\ell}\right) \cdot k}\\ \mathbf{if}\;t \leq -1.88 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	23.9
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{k \cdot {t}^{3}}{\ell} \cdot k}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	36.0
Cost	7040

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

Alternative 14
Error	35.9
Cost	7040

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

Reproduce

herbie shell --seed 2023073 
(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))))