Toniolo and Linder, Equation (13)

Average Error: 34.5 → 29.0

Time: 39.8s

Precision: binary64

Cost: 43528

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot t_1\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t_2 \leq 10^{-138}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(n \cdot \left(\left(U - U*\right) \cdot t_1\right) + \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\left(n \cdot \ell\right) \cdot \frac{\sqrt{\left(U* \cdot U\right) \cdot 2}}{Om}\\ \end{array} \]

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 (pow (/ l Om) 2.0))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 (/ (* l l) Om))) (* (* n t_1) (- U U*)))))))
   (if (<= t_2 1e-138)
     (sqrt
      (*
       n
       (* U (* -2.0 (+ (* n (* (- U U*) t_1)) (- (* l (* 2.0 (/ l Om))) t))))))
     (if (<= t_2 2e+152) t_2 (- (* (* n l) (/ (sqrt (* (* U* U) 2.0)) Om)))))))

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 = pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_1) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e-138) {
		tmp = sqrt((n * (U * (-2.0 * ((n * ((U - U_42_) * t_1)) + ((l * (2.0 * (l / Om))) - t))))));
	} else if (t_2 <= 2e+152) {
		tmp = t_2;
	} else {
		tmp = -((n * l) * (sqrt(((U_42_ * U) * 2.0)) / Om));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (l / om) ** 2.0d0
    t_2 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * t_1) * (u - u_42)))))
    if (t_2 <= 1d-138) then
        tmp = sqrt((n * (u * ((-2.0d0) * ((n * ((u - u_42) * t_1)) + ((l * (2.0d0 * (l / om))) - t))))))
    else if (t_2 <= 2d+152) then
        tmp = t_2
    else
        tmp = -((n * l) * (sqrt(((u_42 * u) * 2.0d0)) / om))
    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.pow((l / Om), 2.0);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_1) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e-138) {
		tmp = Math.sqrt((n * (U * (-2.0 * ((n * ((U - U_42_) * t_1)) + ((l * (2.0 * (l / Om))) - t))))));
	} else if (t_2 <= 2e+152) {
		tmp = t_2;
	} else {
		tmp = -((n * l) * (Math.sqrt(((U_42_ * U) * 2.0)) / Om));
	}
	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.pow((l / Om), 2.0)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_1) * (U - U_42_)))))
	tmp = 0
	if t_2 <= 1e-138:
		tmp = math.sqrt((n * (U * (-2.0 * ((n * ((U - U_42_) * t_1)) + ((l * (2.0 * (l / Om))) - t))))))
	elif t_2 <= 2e+152:
		tmp = t_2
	else:
		tmp = -((n * l) * (math.sqrt(((U_42_ * U) * 2.0)) / Om))
	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 = Float64(l / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * t_1) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 1e-138)
		tmp = sqrt(Float64(n * Float64(U * Float64(-2.0 * Float64(Float64(n * Float64(Float64(U - U_42_) * t_1)) + Float64(Float64(l * Float64(2.0 * Float64(l / Om))) - t))))));
	elseif (t_2 <= 2e+152)
		tmp = t_2;
	else
		tmp = Float64(-Float64(Float64(n * l) * Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) / Om)));
	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 = (l / Om) ^ 2.0;
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_1) * (U - U_42_)))));
	tmp = 0.0;
	if (t_2 <= 1e-138)
		tmp = sqrt((n * (U * (-2.0 * ((n * ((U - U_42_) * t_1)) + ((l * (2.0 * (l / Om))) - t))))));
	elseif (t_2 <= 2e+152)
		tmp = t_2;
	else
		tmp = -((n * l) * (sqrt(((U_42_ * U) * 2.0)) / Om));
	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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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 * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e-138], N[Sqrt[N[(n * N[(U * N[(-2.0 * N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+152], t$95$2, (-N[(N[(n * l), $MachinePrecision] * N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000007e-138

   1. Initial program 50.7

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

   2. Simplified 50.7

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

      [Start] 50.7	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-2 [=>] 50.7	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-44 [=>] 50.7	\[ \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-47 [=>] 50.7	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-2 [=>] 50.7	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational_best-simplify-2 [=>] 50.7	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational_best-simplify-44 [=>] 50.7	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)} \]

   3. Applied egg-rr 36.3

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right) \cdot -2\right)\right)} + 0} \]

   4. Simplified 37.3

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

      [Start] 36.3	\[ \sqrt{U \cdot \left(n \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right) \cdot -2\right)\right)} + 0 \]
      rational_best-simplify-3 [=>] 36.3	\[ \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right) \cdot -2\right)\right)}} \]
      rational_best-simplify-44 [=>] 35.7	\[ \sqrt{\color{blue}{n \cdot \left(U \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right) \cdot -2\right)\right)}} \]
      rational_best-simplify-2 [=>] 35.7	\[ \sqrt{n \cdot \left(U \cdot \color{blue}{\left(-2 \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}\right)} \]
      rational_best-simplify-44 [=>] 37.3	\[ \sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(\color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)\right)} \]
      rational_best-simplify-2 [=>] 37.3	\[ \sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(n \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)\right)} \]
      rational_best-simplify-44 [=>] 37.3	\[ \sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\color{blue}{\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)} - t\right)\right)\right)\right)} \]

   if 1.00000000000000007e-138 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 1.6

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

   if 2.0000000000000001e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 63.6

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

   2. Simplified 55.5

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

      [Start] 63.6	\[ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-2 [=>] 63.6	\[ \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-44 [=>] 63.6	\[ \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-47 [=>] 55.8	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
      rational_best-simplify-2 [=>] 55.8	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)} \]
      rational_best-simplify-2 [=>] 55.8	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\right)} \]
      rational_best-simplify-44 [=>] 55.5	\[ \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U - U*\right) \cdot n\right)}\right)} \]

   3. Taylor expanded in U* around inf 63.0

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

   4. Simplified 61.4

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

      [Start] 63.0	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}} \]
      rational_best-simplify-2 [=>] 63.0	\[ \sqrt{2 \cdot \frac{{n}^{2} \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \]
      rational_best-simplify-44 [=>] 63.1	\[ \sqrt{2 \cdot \frac{\color{blue}{\left(U \cdot U*\right) \cdot \left({n}^{2} \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \]
      rational_best-simplify-47 [=>] 63.1	\[ \sqrt{2 \cdot \color{blue}{\left(\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \frac{U \cdot U*}{{Om}^{2}}\right)}} \]
      exponential-simplify-27 [=>] 61.4	\[ \sqrt{2 \cdot \left(\color{blue}{{\left(\ell \cdot n\right)}^{2}} \cdot \frac{U \cdot U*}{{Om}^{2}}\right)} \]
      rational_best-simplify-2 [<=] 61.4	\[ \sqrt{2 \cdot \left({\color{blue}{\left(n \cdot \ell\right)}}^{2} \cdot \frac{U \cdot U*}{{Om}^{2}}\right)} \]

   5. Taylor expanded in n around -inf 55.8

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

   6. Simplified 55.8

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

      [Start] 55.8	\[ -1 \cdot \left(\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \sqrt{U \cdot U*}\right) \]
      rational_best-simplify-44 [=>] 55.8	\[ \color{blue}{\frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(-1 \cdot \sqrt{U \cdot U*}\right)} \]
      rational_best-simplify-2 [<=] 55.8	\[ \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(-1 \cdot \sqrt{\color{blue}{U* \cdot U}}\right) \]

   7. Applied egg-rr 55.5

      \[\leadsto \color{blue}{-\left(n \cdot \ell\right) \cdot \frac{\sqrt{\left(U* \cdot U\right) \cdot 2}}{Om}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 10^{-138}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(n \cdot \ell\right) \cdot \frac{\sqrt{\left(U* \cdot U\right) \cdot 2}}{Om}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.0
Cost	14884

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ t_2 := \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(-1 \cdot \sqrt{U* \cdot U}\right)\\ t_3 := \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - 2 \cdot t_1\right)}\\ t_4 := U \cdot \left(U* - U\right)\\ \mathbf{if}\;Om \leq -3.6 \cdot 10^{-42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq -3.75 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2 \cdot t_4} \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq -1.7 \cdot 10^{-253}:\\ \;\;\;\;n \cdot \left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{-Om}\right)\\ \mathbf{elif}\;Om \leq -5.8 \cdot 10^{-267}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{-4}{Om}}\\ \mathbf{elif}\;Om \leq 1.45 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{t_4} \cdot \left(\sqrt{2} \cdot \frac{n \cdot \ell}{Om}\right)\\ \mathbf{elif}\;Om \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1080000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + \left(t + -4 \cdot t_1\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	39.9
Cost	14884

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ t_2 := \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - 2 \cdot t_1\right)}\\ t_3 := \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(-1 \cdot \sqrt{U* \cdot U}\right)\\ t_4 := U \cdot \left(U* - U\right)\\ \mathbf{if}\;Om \leq -3.6 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -3.1 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2 \cdot t_4} \cdot \frac{-n}{Om}\right)\\ \mathbf{elif}\;Om \leq -1.8 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq -7 \cdot 10^{-239}:\\ \;\;\;\;n \cdot \left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{-Om}\right)\\ \mathbf{elif}\;Om \leq -6.4 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right) + \left(U \cdot {\ell}^{2}\right) \cdot \left(-4 \cdot \frac{n}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.2 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{t_4} \cdot \left(\sqrt{2} \cdot \frac{n \cdot \ell}{Om}\right)\\ \mathbf{elif}\;Om \leq 7.2 \cdot 10^{-157}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;Om \leq 1080000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + \left(t + -4 \cdot t_1\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	30.0
Cost	14728

\[\begin{array}{l} t_1 := 2 \cdot \frac{1}{Om}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{{Om}^{2}} - t_1\right)\right)} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + \ell \cdot \left(\ell \cdot \frac{-2}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\left(U* \cdot \frac{n}{{Om}^{2}} - t_1\right) \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 4
Error	30.0
Cost	14728

\[\begin{array}{l} t_1 := 2 \cdot \frac{1}{Om}\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(n \cdot \frac{U* - U}{{Om}^{2}} - t_1\right)\right)} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\left(U* \cdot \frac{n}{{Om}^{2}} - t_1\right) \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 5
Error	33.1
Cost	14224

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{-4}{Om}\right)} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Alternative 6
Error	33.1
Cost	14224

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{-4}{Om}\right)} \cdot \left(-\ell\right)\\ \mathbf{elif}\;\ell \leq -2.95 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(t - 2 \cdot t_1\right) \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(t + \left(t + -4 \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Alternative 7
Error	41.1
Cost	14160

\[\begin{array}{l} t_1 := \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(-1 \cdot \sqrt{U* \cdot U}\right)\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;Om \leq -2 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 7.4 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1080000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	41.2
Cost	14096

\[\begin{array}{l} t_1 := -\left(n \cdot \ell\right) \cdot \left(\sqrt{U* \cdot U} \cdot \frac{\sqrt{2}}{Om}\right)\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;Om \leq -6.6 \cdot 10^{-211}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1080000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	41.0
Cost	14096

\[\begin{array}{l} t_1 := \frac{\sqrt{U* \cdot U} \cdot \left(\sqrt{2} \cdot \left(n \cdot \left(-\ell\right)\right)\right)}{Om}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;Om \leq -7 \cdot 10^{-211}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1080000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	41.2
Cost	7696

\[\begin{array}{l} t_1 := -\left(n \cdot \ell\right) \cdot \frac{\sqrt{\left(U* \cdot U\right) \cdot 2}}{Om}\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;Om \leq -1.2 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.1 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 1080000000000:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	36.2
Cost	7240

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;-\ell \cdot \sqrt{U \cdot \left(-4 \cdot \frac{n}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{-4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Alternative 12
Error	39.1
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;U \leq -1.75 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	39.0
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;U \leq -1.75 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 5.4 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	38.6
Cost	7112

\[\begin{array}{l} \mathbf{if}\;U \leq -1.7 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;U \leq 3.4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U + U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \end{array} \]

Alternative 15
Error	40.1
Cost	6848

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

Reproduce

herbie shell --seed 2023096 
(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*))))))