Toniolo and Linder, Equation (13)

Average Error: 34.2 → 32.3

Time: 37.4s

Precision: binary64

Cost: 14608

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

↓

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{n \cdot U}{Om} \cdot -4} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (+ (* 2.0 (* n (* t U))) (* -4.0 (/ (* n (* (pow l 2.0) U)) Om))))))
   (if (<= l -1.2e+154)
     (* (sqrt (* (/ (* n U) Om) -4.0)) (- l))
     (if (<= l -4.2e-90)
       t_1
       (if (<= l -3.5e-275)
         (sqrt (* (* n U) (* t 2.0)))
         (if (<= l 8e+141)
           t_1
           (*
            l
            (sqrt
             (*
              (* n (* U (- (/ (* n U*) (pow Om 2.0)) (* 2.0 (/ 1.0 Om)))))
              2.0)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * (t * U))) + (-4.0 * ((n * (pow(l, 2.0) * U)) / Om))));
	double tmp;
	if (l <= -1.2e+154) {
		tmp = sqrt((((n * U) / Om) * -4.0)) * -l;
	} else if (l <= -4.2e-90) {
		tmp = t_1;
	} else if (l <= -3.5e-275) {
		tmp = sqrt(((n * U) * (t * 2.0)));
	} else if (l <= 8e+141) {
		tmp = t_1;
	} else {
		tmp = l * sqrt(((n * (U * (((n * U_42_) / pow(Om, 2.0)) - (2.0 * (1.0 / Om))))) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

↓

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * (t * u))) + ((-4.0d0) * ((n * ((l ** 2.0d0) * u)) / om))))
    if (l <= (-1.2d+154)) then
        tmp = sqrt((((n * u) / om) * (-4.0d0))) * -l
    else if (l <= (-4.2d-90)) then
        tmp = t_1
    else if (l <= (-3.5d-275)) then
        tmp = sqrt(((n * u) * (t * 2.0d0)))
    else if (l <= 8d+141) then
        tmp = t_1
    else
        tmp = l * sqrt(((n * (u * (((n * u_42) / (om ** 2.0d0)) - (2.0d0 * (1.0d0 / om))))) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * (t * U))) + (-4.0 * ((n * (Math.pow(l, 2.0) * U)) / Om))));
	double tmp;
	if (l <= -1.2e+154) {
		tmp = Math.sqrt((((n * U) / Om) * -4.0)) * -l;
	} else if (l <= -4.2e-90) {
		tmp = t_1;
	} else if (l <= -3.5e-275) {
		tmp = Math.sqrt(((n * U) * (t * 2.0)));
	} else if (l <= 8e+141) {
		tmp = t_1;
	} else {
		tmp = l * Math.sqrt(((n * (U * (((n * U_42_) / Math.pow(Om, 2.0)) - (2.0 * (1.0 / Om))))) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

↓

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * (t * U))) + (-4.0 * ((n * (math.pow(l, 2.0) * U)) / Om))))
	tmp = 0
	if l <= -1.2e+154:
		tmp = math.sqrt((((n * U) / Om) * -4.0)) * -l
	elif l <= -4.2e-90:
		tmp = t_1
	elif l <= -3.5e-275:
		tmp = math.sqrt(((n * U) * (t * 2.0)))
	elif l <= 8e+141:
		tmp = t_1
	else:
		tmp = l * math.sqrt(((n * (U * (((n * U_42_) / math.pow(Om, 2.0)) - (2.0 * (1.0 / Om))))) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

↓

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * Float64(t * U))) + Float64(-4.0 * Float64(Float64(n * Float64((l ^ 2.0) * U)) / Om))))
	tmp = 0.0
	if (l <= -1.2e+154)
		tmp = Float64(sqrt(Float64(Float64(Float64(n * U) / Om) * -4.0)) * Float64(-l));
	elseif (l <= -4.2e-90)
		tmp = t_1;
	elseif (l <= -3.5e-275)
		tmp = sqrt(Float64(Float64(n * U) * Float64(t * 2.0)));
	elseif (l <= 8e+141)
		tmp = t_1;
	else
		tmp = Float64(l * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / (Om ^ 2.0)) - Float64(2.0 * Float64(1.0 / Om))))) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

↓

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * (t * U))) + (-4.0 * ((n * ((l ^ 2.0) * U)) / Om))));
	tmp = 0.0;
	if (l <= -1.2e+154)
		tmp = sqrt((((n * U) / Om) * -4.0)) * -l;
	elseif (l <= -4.2e-90)
		tmp = t_1;
	elseif (l <= -3.5e-275)
		tmp = sqrt(((n * U) * (t * 2.0)));
	elseif (l <= 8e+141)
		tmp = t_1;
	else
		tmp = l * sqrt(((n * (U * (((n * U_42_) / (Om ^ 2.0)) - (2.0 * (1.0 / Om))))) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(n * N[(N[Power[l, 2.0], $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.2e+154], N[(N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * (-l)), $MachinePrecision], If[LessEqual[l, -4.2e-90], t$95$1, If[LessEqual[l, -3.5e-275], N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8e+141], t$95$1, N[(l * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.20000000000000007e154

   1. Initial program 63.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 63.8

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}} \]
      Proof

      [Start] 63.8	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 63.8	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 63.8	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 63.8	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 63.8	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 63.8	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Taylor expanded in Om around inf 63.9

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   4. Taylor expanded in l around -inf 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{n \cdot U}{Om}} \cdot \left(\sqrt{-4} \cdot \ell\right)\right)} \]

   5. Simplified 37.8

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot U}{Om} \cdot -4} \cdot \left(-\ell\right)} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\sqrt{\frac{n \cdot U}{Om}} \cdot \left(\sqrt{-4} \cdot \ell\right)\right) \]
      rational.json-simplify-2 [<=] 64.0	\[ -1 \cdot \left(\sqrt{\frac{n \cdot U}{Om}} \cdot \color{blue}{\left(\ell \cdot \sqrt{-4}\right)}\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ -1 \cdot \color{blue}{\left(\ell \cdot \left(\sqrt{-4} \cdot \sqrt{\frac{n \cdot U}{Om}}\right)\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ -1 \cdot \color{blue}{\left(\left(\sqrt{-4} \cdot \sqrt{\frac{n \cdot U}{Om}}\right) \cdot \ell\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{\left(\sqrt{-4} \cdot \sqrt{\frac{n \cdot U}{Om}}\right) \cdot \left(\ell \cdot -1\right)} \]
      exponential.json-simplify-20 [=>] 37.8	\[ \color{blue}{\sqrt{\frac{n \cdot U}{Om} \cdot -4}} \cdot \left(\ell \cdot -1\right) \]
      rational.json-simplify-9 [=>] 37.8	\[ \sqrt{\frac{n \cdot U}{Om} \cdot -4} \cdot \color{blue}{\left(-\ell\right)} \]

   if -1.20000000000000007e154 < l < -4.1999999999999998e-90 or -3.49999999999999969e-275 < l < 8.00000000000000014e141

   1. Initial program 28.1

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 29.7

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}} \]
      Proof

      [Start] 28.1	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 28.1	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 28.1	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 28.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 28.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 29.7	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Taylor expanded in Om around inf 32.1

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   if -4.1999999999999998e-90 < l < -3.49999999999999969e-275

   1. Initial program 25.5

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 28.1

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}} \]
      Proof

      [Start] 25.5	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 25.5	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 25.5	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 25.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 25.5	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 28.1	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Taylor expanded in Om around inf 30.2

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}} \]

   4. Taylor expanded in l around 0 30.3

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)}} \]

   5. Simplified 29.6

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}} \]
      Proof

      [Start] 30.3	\[ \sqrt{2} \cdot \sqrt{n \cdot \left(t \cdot U\right)} \]
      rational.json-simplify-43 [<=] 30.8	\[ \sqrt{2} \cdot \sqrt{\color{blue}{U \cdot \left(n \cdot t\right)}} \]
      exponential.json-simplify-20 [=>] 30.7	\[ \color{blue}{\sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]
      rational.json-simplify-2 [<=] 30.7	\[ \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
      rational.json-simplify-2 [=>] 30.7	\[ \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(t \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 29.6	\[ \sqrt{2 \cdot \color{blue}{\left(t \cdot \left(n \cdot U\right)\right)}} \]
      rational.json-simplify-2 [=>] 29.6	\[ \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]
      rational.json-simplify-43 [=>] 29.6	\[ \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}} \]

   if 8.00000000000000014e141 < l

   1. Initial program 62.0

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Simplified 62.0

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}} \]
      Proof

      [Start] 62.0	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 62.0	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-43 [=>] 62.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational.json-simplify-2 [=>] 62.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational.json-simplify-2 [=>] 62.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational.json-simplify-43 [=>] 62.0	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)} \]

   3. Taylor expanded in U around 0 62.9

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}}\right)} \]

   4. Simplified 62.9

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(-\frac{n \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}\right)}\right)} \]
      Proof

      [Start] 62.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - -1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)} \]
      rational.json-simplify-2 [=>] 62.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} \cdot -1}\right)} \]
      rational.json-simplify-9 [=>] 62.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(-\frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}}\right)}\right)} \]
      rational.json-simplify-2 [=>] 62.9	\[ \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(-\frac{n \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}\right)\right)} \]

   5. Taylor expanded in l around inf 34.4

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}} \]

   6. Simplified 34.3

      \[\leadsto \color{blue}{\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot 2}} \]
      Proof

      [Start] 34.4	\[ \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \]
      rational.json-simplify-2 [=>] 34.4	\[ \color{blue}{\sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \cdot \left(\sqrt{2} \cdot \ell\right)} \]
      rational.json-simplify-2 [=>] 34.4	\[ \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)} \cdot \color{blue}{\left(\ell \cdot \sqrt{2}\right)} \]
      rational.json-simplify-43 [=>] 34.4	\[ \color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)}\right)} \]
      exponential.json-simplify-20 [=>] 34.3	\[ \ell \cdot \color{blue}{\sqrt{\left(n \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right) \cdot 2}} \]
      rational.json-simplify-2 [=>] 34.3	\[ \ell \cdot \sqrt{\left(n \cdot \color{blue}{\left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right) \cdot 2} \]

3. Recombined 4 regimes into one program.

4. Final simplification 32.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{n \cdot U}{Om} \cdot -4} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right) \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Error	32.8
Cost	14480

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\\ t_2 := \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \mathbf{if}\;\ell \leq -1.12 \cdot 10^{+154}:\\ \;\;\;\;t_2 \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot t_2\\ \end{array} \]

Alternative 2
Error	32.4
Cost	14344

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{n \cdot U}{Om} \cdot -4} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right) + -4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right) \cdot 2}\\ \end{array} \]

Alternative 3
Error	32.6
Cost	14224

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)\right)}\\ t_2 := \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+126}:\\ \;\;\;\;t_2 \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -2.2 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot t_2\\ \end{array} \]

Alternative 4
Error	32.8
Cost	13960

\[\begin{array}{l} t_1 := \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+125}:\\ \;\;\;\;t_1 \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot t_1\\ \end{array} \]

Alternative 5
Error	35.7
Cost	7240

\[\begin{array}{l} t_1 := \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+101}:\\ \;\;\;\;t_1 \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot t_1\\ \end{array} \]

Alternative 6
Error	37.9
Cost	7108

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\frac{n \cdot U}{Om} \cdot -4}\\ \end{array} \]

Alternative 7
Error	39.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;U \leq 1.55 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t + t\right)\right)}\\ \end{array} \]

Alternative 8
Error	39.9
Cost	6848

\[\sqrt{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)} \]

Reproduce

herbie shell --seed 2023077 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))