Toniolo and Linder, Equation (10+)

Average Error: 33.0 → 17.2

Time: 32.8s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\cos k}{{\sin k}^{2}}}{t \cdot {k}^{2}} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (*
          2.0
          (/
           (/ (/ l (tan k)) (pow t 3.0))
           (* (+ 2.0 (pow (/ k t) 2.0)) (/ (sin k) l))))))
   (if (<= t -6.4e-68)
     t_1
     (if (<= t 1.6e-47)
       (* 2.0 (* (/ (* l (/ (cos k) (pow (sin k) 2.0))) (* t (pow k 2.0))) l))
       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 = 2.0 * (((l / tan(k)) / pow(t, 3.0)) / ((2.0 + pow((k / t), 2.0)) * (sin(k) / l)));
	double tmp;
	if (t <= -6.4e-68) {
		tmp = t_1;
	} else if (t <= 1.6e-47) {
		tmp = 2.0 * (((l * (cos(k) / pow(sin(k), 2.0))) / (t * pow(k, 2.0))) * l);
	} 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 = 2.0d0 * (((l / tan(k)) / (t ** 3.0d0)) / ((2.0d0 + ((k / t) ** 2.0d0)) * (sin(k) / l)))
    if (t <= (-6.4d-68)) then
        tmp = t_1
    else if (t <= 1.6d-47) then
        tmp = 2.0d0 * (((l * (cos(k) / (sin(k) ** 2.0d0))) / (t * (k ** 2.0d0))) * l)
    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 = 2.0 * (((l / Math.tan(k)) / Math.pow(t, 3.0)) / ((2.0 + Math.pow((k / t), 2.0)) * (Math.sin(k) / l)));
	double tmp;
	if (t <= -6.4e-68) {
		tmp = t_1;
	} else if (t <= 1.6e-47) {
		tmp = 2.0 * (((l * (Math.cos(k) / Math.pow(Math.sin(k), 2.0))) / (t * Math.pow(k, 2.0))) * l);
	} 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 = 2.0 * (((l / math.tan(k)) / math.pow(t, 3.0)) / ((2.0 + math.pow((k / t), 2.0)) * (math.sin(k) / l)))
	tmp = 0
	if t <= -6.4e-68:
		tmp = t_1
	elif t <= 1.6e-47:
		tmp = 2.0 * (((l * (math.cos(k) / math.pow(math.sin(k), 2.0))) / (t * math.pow(k, 2.0))) * l)
	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(2.0 * Float64(Float64(Float64(l / tan(k)) / (t ^ 3.0)) / Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(sin(k) / l))))
	tmp = 0.0
	if (t <= -6.4e-68)
		tmp = t_1;
	elseif (t <= 1.6e-47)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(cos(k) / (sin(k) ^ 2.0))) / Float64(t * (k ^ 2.0))) * l));
	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 = 2.0 * (((l / tan(k)) / (t ^ 3.0)) / ((2.0 + ((k / t) ^ 2.0)) * (sin(k) / l)));
	tmp = 0.0;
	if (t <= -6.4e-68)
		tmp = t_1;
	elseif (t <= 1.6e-47)
		tmp = 2.0 * (((l * (cos(k) / (sin(k) ^ 2.0))) / (t * (k ^ 2.0))) * l);
	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[(2.0 * N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.4e-68], t$95$1, If[LessEqual[t, 1.6e-47], N[(2.0 * N[(N[(N[(l * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.3999999999999998e-68 or 1.6e-47 < t

   1. Initial program 22.8

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

   2. Simplified 17.8

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

      [Start] 22.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 22.8	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-46 [=>] 22.8	\[ \frac{\color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 22.8	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 22.8	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 22.0	\[ \frac{\frac{\color{blue}{\frac{\ell \cdot \ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-49 [=>] 17.8	\[ \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 17.8	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 17.8	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 17.8	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 17.8	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 17.8	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 17.8

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right)\right) \cdot \frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)}} \]

   4. Simplified 16.5

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

      [Start] 17.8	\[ \left(\ell \cdot \left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right)\right) \cdot \frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \]
      rational.json-simplify-2 [<=] 17.8	\[ \color{blue}{\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \left(\ell \cdot \left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right)\right)} \]
      rational.json-simplify-43 [<=] 16.8	\[ \color{blue}{\left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right)} \]
      metadata-eval [<=] 16.8	\[ \left(\color{blue}{\left(1 + 1\right)} \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-7 [<=] 16.8	\[ \left(\left(1 + 1\right) \cdot \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}{1}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-30 [=>] 16.8	\[ \color{blue}{\left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} + \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}{1}\right)} \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-7 [=>] 16.8	\[ \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} + \color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-53 [<=] 17.6	\[ \color{blue}{\left(\ell + \ell\right) \cdot \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} \cdot \frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)}\right)} \]
      rational.json-simplify-43 [<=] 17.6	\[ \left(\ell + \ell\right) \cdot \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} \cdot \frac{2}{\color{blue}{4 \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}\right) \]
      rational.json-simplify-46 [=>] 17.6	\[ \left(\ell + \ell\right) \cdot \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} \cdot \color{blue}{\frac{\frac{2}{4}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \]

   5. Applied egg-rr 16.4

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

   6. Simplified 15.5

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

      [Start] 16.4	\[ 2 \cdot \frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\sin k}{\ell}\right)} \]
      rational.json-simplify-43 [=>] 16.4	\[ 2 \cdot \frac{\ell}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      rational.json-simplify-46 [=>] 15.9	\[ 2 \cdot \color{blue}{\frac{\frac{\ell}{\tan k \cdot {t}^{3}}}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-46 [=>] 15.5	\[ 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      rational.json-simplify-2 [=>] 15.5	\[ 2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}} \]

   if -6.3999999999999998e-68 < t < 1.6e-47

   1. Initial program 57.2

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

   2. Simplified 57.5

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

      [Start] 57.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 57.4	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-46 [=>] 57.4	\[ \frac{\color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 57.4	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 57.3	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 57.8	\[ \frac{\frac{\color{blue}{\frac{\ell \cdot \ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-49 [=>] 57.5	\[ \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 57.5	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 57.5	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 57.5	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 57.5	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 57.5	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 55.3

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right)\right) \cdot \frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)}} \]

   4. Simplified 57.1

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

      [Start] 55.3	\[ \left(\ell \cdot \left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right)\right) \cdot \frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \]
      rational.json-simplify-2 [<=] 55.3	\[ \color{blue}{\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \left(\ell \cdot \left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right)\right)} \]
      rational.json-simplify-43 [<=] 55.1	\[ \color{blue}{\left(2 \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right)} \]
      metadata-eval [<=] 55.1	\[ \left(\color{blue}{\left(1 + 1\right)} \cdot \frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-7 [<=] 55.1	\[ \left(\left(1 + 1\right) \cdot \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}{1}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-30 [=>] 55.1	\[ \color{blue}{\left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} + \frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}{1}\right)} \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-7 [=>] 55.1	\[ \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} + \color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}\right) \cdot \left(\frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)} \cdot \ell\right) \]
      rational.json-simplify-53 [<=] 56.1	\[ \color{blue}{\left(\ell + \ell\right) \cdot \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} \cdot \frac{2}{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 4\right)}\right)} \]
      rational.json-simplify-43 [<=] 56.1	\[ \left(\ell + \ell\right) \cdot \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} \cdot \frac{2}{\color{blue}{4 \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}\right) \]
      rational.json-simplify-46 [=>] 56.1	\[ \left(\ell + \ell\right) \cdot \left(\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}} \cdot \color{blue}{\frac{\frac{2}{4}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \]

   5. Taylor expanded in k around inf 22.9

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

   6. Simplified 21.3

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

      [Start] 22.9	\[ 2 \cdot \left(\frac{\cos k \cdot \ell}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \ell\right) \]
      rational.json-simplify-43 [=>] 22.9	\[ 2 \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}} \cdot \ell\right) \]
      rational.json-simplify-46 [=>] 21.3	\[ 2 \cdot \left(\color{blue}{\frac{\frac{\cos k \cdot \ell}{{\sin k}^{2}}}{t \cdot {k}^{2}}} \cdot \ell\right) \]
      rational.json-simplify-49 [=>] 21.3	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \frac{\cos k}{{\sin k}^{2}}}}{t \cdot {k}^{2}} \cdot \ell\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-68}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\cos k}{{\sin k}^{2}}}{t \cdot {k}^{2}} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.3
Cost	26824

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell \cdot \frac{\cos k}{{\sin k}^{2}}}{t \cdot {k}^{2}} \cdot \ell\right)\\ \mathbf{if}\;k \leq -1.72 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 0.35:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot {t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.3
Cost	26824

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.72 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\cos k}{t_1}}{t \cdot {k}^{2}} \cdot \ell\right)\\ \mathbf{elif}\;k \leq 0.95:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot {t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \ell}{{k}^{2} \cdot \left(t_1 \cdot t\right)} \cdot \ell\right)\\ \end{array} \]

Alternative 3
Error	20.6
Cost	20680

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\cos k \cdot \frac{\ell}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right) \cdot \ell\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 4200:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{k \cdot {t}^{3}} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	20.3
Cost	20680

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\cos k \cdot \frac{\ell}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right) \cdot \ell\right)\\ \mathbf{if}\;k \leq -5 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 0.135:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot {t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	19.8
Cost	20360

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\cos k \cdot \frac{\ell}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right) \cdot \ell\right)\\ \mathbf{if}\;k \leq -5 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 70000:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{2 \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.9
Cost	20360

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\cos k \cdot \frac{\ell}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right) \cdot \ell\right)\\ \mathbf{if}\;k \leq -4.9 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7000000000:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{2 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	26.0
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{2 \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 8
Error	25.8
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{2 \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{\left(k \cdot \sin k\right)}^{2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 9
Error	26.0
Cost	13828

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\tan k}}{{t}^{3}}}{2 \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 10
Error	26.4
Cost	13704

\[\begin{array}{l} t_1 := \ell \cdot \frac{\frac{\ell}{k}}{\sin k \cdot {t}^{3}}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	26.0
Cost	13704

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	38.4
Cost	13376

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

Alternative 13
Error	38.6
Cost	13376

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

Reproduce

herbie shell --seed 2023068 
(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))))