Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 66.8%

Time: 24.4s

Alternatives: 22

Speedup: 1.8×

Specification

Math

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

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*))))))

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_)))));
}

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

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_)))));
}

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_)))))

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

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

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]

Initial Program: 50.0% accurate, 1.0× speedup

Math

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

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*))))))

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_)))));
}

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

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_)))));
}

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_)))))

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

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

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]

Alternative 1: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(t + \frac{U* \cdot \frac{l\_m}{\frac{Om}{n}} + l\_m \cdot -2}{\frac{Om}{l\_m}}\right)\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (*
      (sqrt
       (*
        U
        (* 2.0 (+ t (/ (+ (* U* (/ l_m (/ Om n))) (* l_m -2.0)) (/ Om l_m))))))
      (sqrt n))
     (if (<= t_1 4e+275)
       (sqrt t_1)
       (*
        (sqrt (/ (* (* n U) (+ -2.0 (/ (* n U*) Om))) Om))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * sqrt(n);
	} else if (t_1 <= 4e+275) {
		tmp = sqrt(t_1);
	} else {
		tmp = sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((u * (2.0d0 * (t + (((u_42 * (l_m / (om / n))) + (l_m * (-2.0d0))) / (om / l_m)))))) * sqrt(n)
    else if (t_1 <= 4d+275) then
        tmp = sqrt(t_1)
    else
        tmp = sqrt((((n * u) * ((-2.0d0) + ((n * u_42) / om))) / om)) * (l_m * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * Math.sqrt(n);
	} else if (t_1 <= 4e+275) {
		tmp = Math.sqrt(t_1);
	} else {
		tmp = Math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * math.sqrt(n)
	elif t_1 <= 4e+275:
		tmp = math.sqrt(t_1)
	else:
		tmp = math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * math.sqrt(2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(U_42_ * Float64(l_m / Float64(Om / n))) + Float64(l_m * -2.0)) / Float64(Om / l_m)))))) * sqrt(n));
	elseif (t_1 <= 4e+275)
		tmp = sqrt(t_1);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(n * U) * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))) / Om)) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * sqrt(n);
	elseif (t_1 <= 4e+275)
		tmp = sqrt(t_1);
	else
		tmp = sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(U * N[(2.0 * N[(t + N[(N[(N[(U$42$ * N[(l$95$m / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+275], N[Sqrt[t$95$1], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 11.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 42.2%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 42.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 42.8%

      \[\leadsto \sqrt{\left(\left(t + \frac{\color{blue}{U* \cdot \left(\ell \cdot \frac{n}{Om}\right)} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot n\right)\right) \cdot U} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{U \cdot \left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot 2\right)}\right), \color{blue}{\left({n}^{\frac{1}{2}}\right)}\right) \]

   11. Applied egg-rr 50.7%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.99999999999999984e275

   1. Initial program 97.6%

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

   if 3.99999999999999984e275 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 16.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 38.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 41.7%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 41.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 41.7%

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

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right) \]

       2. distribute-lft-out N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(U \cdot n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{/.f64}\left(\left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), \left(Om \cdot t\right)\right)\right)\right)\right)\right) \]

   12. Simplified 36.6%

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

   13. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\sqrt{\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}}\right), \color{blue}{\left(\ell \cdot \sqrt{2}\right)}\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}\right)\right), \left(\color{blue}{\ell} \cdot \sqrt{2}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U \cdot n\right), \left(\frac{U* \cdot n}{Om} - 2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{Om} - 2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{Om} + \left(\mathsf{neg}\left(2\right)\right)\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{Om} + -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{U* \cdot n}{Om}\right), -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot n\right), Om\right), -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right), -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right), -2\right)\right), Om\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

       13. sqrt-lowering-sqrt.f64 29.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right), -2\right)\right), Om\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   15. Simplified 29.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(t + \frac{U* \cdot \frac{\ell}{\frac{Om}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 4 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(l\_m \cdot -2 + \frac{n}{\frac{Om}{l\_m}} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \left(\left(l\_m \cdot \left(n \cdot l\_m\right)\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(t + \frac{U* \cdot \frac{l\_m}{\frac{Om}{n}} + l\_m \cdot -2}{\frac{Om}{l\_m}}\right)\right)} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -1.4e-174)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ t (* (/ l_m Om) (+ (* l_m -2.0) (* (/ n (/ Om l_m)) (- U* U)))))))
   (if (<= n 7e-278)
     (sqrt
      (+
       (* (* 2.0 U) (* n t))
       (* (* U -4.0) (* (* l_m (* n l_m)) (/ 1.0 Om)))))
     (*
      (sqrt
       (*
        U
        (* 2.0 (+ t (/ (+ (* U* (/ l_m (/ Om n))) (* l_m -2.0)) (/ Om l_m))))))
      (sqrt n)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -1.4e-174) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + ((n / (Om / l_m)) * (U_42_ - U)))))));
	} else if (n <= 7e-278) {
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	} else {
		tmp = sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * sqrt(n);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-1.4d-174)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l_m / om) * ((l_m * (-2.0d0)) + ((n / (om / l_m)) * (u_42 - u)))))))
    else if (n <= 7d-278) then
        tmp = sqrt((((2.0d0 * u) * (n * t)) + ((u * (-4.0d0)) * ((l_m * (n * l_m)) * (1.0d0 / om)))))
    else
        tmp = sqrt((u * (2.0d0 * (t + (((u_42 * (l_m / (om / n))) + (l_m * (-2.0d0))) / (om / l_m)))))) * sqrt(n)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -1.4e-174) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + ((n / (Om / l_m)) * (U_42_ - U)))))));
	} else if (n <= 7e-278) {
		tmp = Math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	} else {
		tmp = Math.sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * Math.sqrt(n);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -1.4e-174:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + ((n / (Om / l_m)) * (U_42_ - U)))))))
	elif n <= 7e-278:
		tmp = math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))))
	else:
		tmp = math.sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * math.sqrt(n)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -1.4e-174)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l_m / Om) * Float64(Float64(l_m * -2.0) + Float64(Float64(n / Float64(Om / l_m)) * Float64(U_42_ - U)))))));
	elseif (n <= 7e-278)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * Float64(n * t)) + Float64(Float64(U * -4.0) * Float64(Float64(l_m * Float64(n * l_m)) * Float64(1.0 / Om)))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(U_42_ * Float64(l_m / Float64(Om / n))) + Float64(l_m * -2.0)) / Float64(Om / l_m)))))) * sqrt(n));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= -1.4e-174)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + ((n / (Om / l_m)) * (U_42_ - U)))))));
	elseif (n <= 7e-278)
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	else
		tmp = sqrt((U * (2.0 * (t + (((U_42_ * (l_m / (Om / n))) + (l_m * -2.0)) / (Om / l_m)))))) * sqrt(n);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -1.4e-174], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(n / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 7e-278], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(U * -4.0), $MachinePrecision] * N[(N[(l$95$m * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(2.0 * N[(t + N[(N[(N[(U$42$ * N[(l$95$m / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.39999999999999999e-174

   1. Initial program 50.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(\frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{n \cdot \ell}{Om}\right), \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(n \cdot \frac{\ell}{Om}\right), \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(n \cdot \frac{1}{\frac{Om}{\ell}}\right), \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{\frac{Om}{\ell}}\right), \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right), \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right), \left(U* - U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 62.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 62.7%

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

   if -1.39999999999999999e-174 < n < 6.99999999999999941e-278

   1. Initial program 34.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 45.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 61.3%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

      13. *-lowering-*.f64 48.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

   9. Simplified 48.6%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right) \cdot \frac{1}{Om}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(\left(-4 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{Om}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(-4 \cdot U\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot -4\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 69.5%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(1, Om\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 69.5%

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot -4\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{1}{Om}\right)}} \]

   if 6.99999999999999941e-278 < n

   1. Initial program 46.7%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 58.7%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 58.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 58.8%

      \[\leadsto \sqrt{\left(\left(t + \frac{\color{blue}{U* \cdot \left(\ell \cdot \frac{n}{Om}\right)} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot n\right)\right) \cdot U} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{U \cdot \left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot 2\right)}\right), \color{blue}{\left({n}^{\frac{1}{2}}\right)}\right) \]

   11. Applied egg-rr 67.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(t + \frac{U* \cdot \frac{\ell}{\frac{Om}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{l\_m \cdot -2 + U* \cdot \left(l\_m \cdot \frac{n}{Om}\right)}{\frac{Om}{l\_m}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\left(U* - U\right) \cdot \frac{n}{Om}}{Om} + \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.05e+210)
   (sqrt
    (*
     U
     (*
      (* 2.0 n)
      (+ t (/ (+ (* l_m -2.0) (* U* (* l_m (/ n Om)))) (/ Om l_m))))))
   (*
    (* l_m (sqrt 2.0))
    (sqrt (* (* n U) (+ (/ (* (- U* U) (/ n Om)) Om) (/ -2.0 Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+210) {
		tmp = sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((((U_42_ - U) * (n / Om)) / Om) + (-2.0 / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.05d+210) then
        tmp = sqrt((u * ((2.0d0 * n) * (t + (((l_m * (-2.0d0)) + (u_42 * (l_m * (n / om)))) / (om / l_m))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt(((n * u) * ((((u_42 - u) * (n / om)) / om) + ((-2.0d0) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+210) {
		tmp = Math.sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt(((n * U) * ((((U_42_ - U) * (n / Om)) / Om) + (-2.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.05e+210:
		tmp = math.sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt(((n * U) * ((((U_42_ - U) * (n / Om)) / Om) + (-2.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.05e+210)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t + Float64(Float64(Float64(l_m * -2.0) + Float64(U_42_ * Float64(l_m * Float64(n / Om)))) / Float64(Om / l_m))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(Float64(Float64(U_42_ - U) * Float64(n / Om)) / Om) + Float64(-2.0 / Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.05e+210)
		tmp = sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((((U_42_ - U) * (n / Om)) / Om) + (-2.0 / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.05e+210], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(U$42$ * N[(l$95$m * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.05e210

   1. Initial program 48.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 60.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 60.6%

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

   if 2.05e210 < l

   1. Initial program 16.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 38.5%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{U \cdot \color{blue}{\left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot n\right), \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \left(U* - U\right)\right), \left({Om}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U* - U\right)\right), \left({Om}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \left({Om}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \left(Om \cdot Om\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{Om}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\mathsf{neg}\left(\frac{2}{Om}\right)\right)\right)\right)\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{Om}\right)\right)\right)\right)\right) \]

   7. Simplified 79.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{\frac{n}{Om} \cdot \left(U* - U\right)}{Om}\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{n}{Om} \cdot \left(U* - U\right)\right), Om\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{Om}\right), \left(U* - U\right)\right), Om\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, Om\right), \left(U* - U\right)\right), Om\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 86.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, Om\right), \mathsf{\_.f64}\left(U*, U\right)\right), Om\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 86.2%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\left(U* - U\right) \cdot \frac{n}{Om}}{Om} + \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{l\_m \cdot -2 + U* \cdot \left(l\_m \cdot \frac{n}{Om}\right)}{\frac{Om}{l\_m}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.05e+210)
   (sqrt
    (*
     U
     (*
      (* 2.0 n)
      (+ t (/ (+ (* l_m -2.0) (* U* (* l_m (/ n Om)))) (/ Om l_m))))))
   (* (sqrt (/ (* (* n U) (+ -2.0 (/ (* n U*) Om))) Om)) (* l_m (sqrt 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+210) {
		tmp = sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	} else {
		tmp = sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.05d+210) then
        tmp = sqrt((u * ((2.0d0 * n) * (t + (((l_m * (-2.0d0)) + (u_42 * (l_m * (n / om)))) / (om / l_m))))))
    else
        tmp = sqrt((((n * u) * ((-2.0d0) + ((n * u_42) / om))) / om)) * (l_m * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+210) {
		tmp = Math.sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	} else {
		tmp = Math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.05e+210:
		tmp = math.sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))))
	else:
		tmp = math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * math.sqrt(2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.05e+210)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t + Float64(Float64(Float64(l_m * -2.0) + Float64(U_42_ * Float64(l_m * Float64(n / Om)))) / Float64(Om / l_m))))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(n * U) * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))) / Om)) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.05e+210)
		tmp = sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	else
		tmp = sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om)) * (l_m * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.05e+210], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(U$42$ * N[(l$95$m * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.05e210

   1. Initial program 48.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 60.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 60.6%

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

   if 2.05e210 < l

   1. Initial program 16.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 38.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 38.8%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 39.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 39.3%

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

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right) \]

       2. distribute-lft-out N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(U \cdot n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{/.f64}\left(\left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), \left(Om \cdot t\right)\right)\right)\right)\right)\right) \]

   12. Simplified 37.7%

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

   13. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\sqrt{\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}}\right), \color{blue}{\left(\ell \cdot \sqrt{2}\right)}\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}\right)\right), \left(\color{blue}{\ell} \cdot \sqrt{2}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U \cdot n\right), \left(\frac{U* \cdot n}{Om} - 2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{Om} - 2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{Om} + \left(\mathsf{neg}\left(2\right)\right)\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{Om} + -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{U* \cdot n}{Om}\right), -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot n\right), Om\right), -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right), -2\right)\right), Om\right)\right), \left(\ell \cdot \sqrt{2}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right), -2\right)\right), Om\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

       13. sqrt-lowering-sqrt.f64 79.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right), -2\right)\right), Om\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   15. Simplified 79.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(l\_m \cdot -2 + U* \cdot \frac{n \cdot l\_m}{Om}\right)\right)}\\ \mathbf{if}\;n \leq -1.65 \cdot 10^{-174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \left(\left(l\_m \cdot \left(n \cdot l\_m\right)\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 (* n U))
           (+ t (* (/ l_m Om) (+ (* l_m -2.0) (* U* (/ (* n l_m) Om)))))))))
   (if (<= n -1.65e-174)
     t_1
     (if (<= n 1.8e-67)
       (sqrt
        (+
         (* (* 2.0 U) (* n t))
         (* (* U -4.0) (* (* l_m (* n l_m)) (/ 1.0 Om)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om)))))));
	double tmp;
	if (n <= -1.65e-174) {
		tmp = t_1;
	} else if (n <= 1.8e-67) {
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * (t + ((l_m / om) * ((l_m * (-2.0d0)) + (u_42 * ((n * l_m) / om)))))))
    if (n <= (-1.65d-174)) then
        tmp = t_1
    else if (n <= 1.8d-67) then
        tmp = sqrt((((2.0d0 * u) * (n * t)) + ((u * (-4.0d0)) * ((l_m * (n * l_m)) * (1.0d0 / om)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om)))))));
	double tmp;
	if (n <= -1.65e-174) {
		tmp = t_1;
	} else if (n <= 1.8e-67) {
		tmp = Math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om)))))))
	tmp = 0
	if n <= -1.65e-174:
		tmp = t_1
	elif n <= 1.8e-67:
		tmp = math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l_m / Om) * Float64(Float64(l_m * -2.0) + Float64(U_42_ * Float64(Float64(n * l_m) / Om)))))))
	tmp = 0.0
	if (n <= -1.65e-174)
		tmp = t_1;
	elseif (n <= 1.8e-67)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * Float64(n * t)) + Float64(Float64(U * -4.0) * Float64(Float64(l_m * Float64(n * l_m)) * Float64(1.0 / Om)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om)))))));
	tmp = 0.0;
	if (n <= -1.65e-174)
		tmp = t_1;
	elseif (n <= 1.8e-67)
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.65e-174], t$95$1, If[LessEqual[n, 1.8e-67], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(U * -4.0), $MachinePrecision] * N[(N[(l$95$m * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.65e-174 or 1.8e-67 < n

   1. Initial program 51.0%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 62.8%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(n \cdot \ell\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 62.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \ell\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right)\right)\right) \]

   7. Simplified 62.2%

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

   if -1.65e-174 < n < 1.8e-67

   1. Initial program 38.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 46.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

      13. *-lowering-*.f64 47.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

   9. Simplified 47.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right) \cdot \frac{1}{Om}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(\left(-4 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{Om}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(-4 \cdot U\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot -4\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 62.8%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(1, Om\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 62.8%

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot -4\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{1}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}\right)}\\ \mathbf{if}\;n \leq -3.05 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \left(\left(l\_m \cdot \left(n \cdot l\_m\right)\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (* 2.0 (* n U)) (+ t (/ (* U* (* n (* l_m l_m))) (* Om Om)))))))
   (if (<= n -3.05e+126)
     t_1
     (if (<= n 4.1e-67)
       (sqrt
        (+
         (* (* 2.0 U) (* n t))
         (* (* U -4.0) (* (* l_m (* n l_m)) (/ 1.0 Om)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	double tmp;
	if (n <= -3.05e+126) {
		tmp = t_1;
	} else if (n <= 4.1e-67) {
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * (t + ((u_42 * (n * (l_m * l_m))) / (om * om)))))
    if (n <= (-3.05d+126)) then
        tmp = t_1
    else if (n <= 4.1d-67) then
        tmp = sqrt((((2.0d0 * u) * (n * t)) + ((u * (-4.0d0)) * ((l_m * (n * l_m)) * (1.0d0 / om)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	double tmp;
	if (n <= -3.05e+126) {
		tmp = t_1;
	} else if (n <= 4.1e-67) {
		tmp = Math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))))
	tmp = 0
	if n <= -3.05e+126:
		tmp = t_1
	elif n <= 4.1e-67:
		tmp = math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / Float64(Om * Om)))))
	tmp = 0.0
	if (n <= -3.05e+126)
		tmp = t_1;
	elseif (n <= 4.1e-67)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * Float64(n * t)) + Float64(Float64(U * -4.0) * Float64(Float64(l_m * Float64(n * l_m)) * Float64(1.0 / Om)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	tmp = 0.0;
	if (n <= -3.05e+126)
		tmp = t_1;
	elseif (n <= 4.1e-67)
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) * (1.0 / Om)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3.05e+126], t$95$1, If[LessEqual[n, 4.1e-67], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(U * -4.0), $MachinePrecision] * N[(N[(l$95$m * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.05000000000000011e126 or 4.0999999999999997e-67 < n

   1. Initial program 47.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 64.7%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(n \cdot {\ell}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right) \]

   7. Simplified 56.2%

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

   if -3.05000000000000011e126 < n < 4.0999999999999997e-67

   1. Initial program 45.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 58.9%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

      13. *-lowering-*.f64 48.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

   9. Simplified 48.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right) \cdot \frac{1}{Om}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(\left(-4 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{Om}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(-4 \cdot U\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot -4\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{1}{Om}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \left(\frac{1}{Om}\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 58.1%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(1, Om\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 58.1%

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot -4\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \frac{1}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.05 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}\right)}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \frac{l\_m \cdot \left(n \cdot l\_m\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (* 2.0 (* n U)) (+ t (/ (* U* (* n (* l_m l_m))) (* Om Om)))))))
   (if (<= n -3.4e+126)
     t_1
     (if (<= n 9.8e-67)
       (sqrt (+ (* (* 2.0 U) (* n t)) (* (* U -4.0) (/ (* l_m (* n l_m)) Om))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	double tmp;
	if (n <= -3.4e+126) {
		tmp = t_1;
	} else if (n <= 9.8e-67) {
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * (t + ((u_42 * (n * (l_m * l_m))) / (om * om)))))
    if (n <= (-3.4d+126)) then
        tmp = t_1
    else if (n <= 9.8d-67) then
        tmp = sqrt((((2.0d0 * u) * (n * t)) + ((u * (-4.0d0)) * ((l_m * (n * l_m)) / om))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	double tmp;
	if (n <= -3.4e+126) {
		tmp = t_1;
	} else if (n <= 9.8e-67) {
		tmp = Math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))))
	tmp = 0
	if n <= -3.4e+126:
		tmp = t_1
	elif n <= 9.8e-67:
		tmp = math.sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) / Om))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / Float64(Om * Om)))))
	tmp = 0.0
	if (n <= -3.4e+126)
		tmp = t_1;
	elseif (n <= 9.8e-67)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * Float64(n * t)) + Float64(Float64(U * -4.0) * Float64(Float64(l_m * Float64(n * l_m)) / Om))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t + ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	tmp = 0.0;
	if (n <= -3.4e+126)
		tmp = t_1;
	elseif (n <= 9.8e-67)
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * -4.0) * ((l_m * (n * l_m)) / Om))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3.4e+126], t$95$1, If[LessEqual[n, 9.8e-67], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(U * -4.0), $MachinePrecision] * N[(N[(l$95$m * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.39999999999999989e126 or 9.79999999999999987e-67 < n

   1. Initial program 47.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 64.7%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(n \cdot {\ell}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right) \]

   7. Simplified 56.2%

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

   if -3.39999999999999989e126 < n < 9.79999999999999987e-67

   1. Initial program 45.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 58.9%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

      13. *-lowering-*.f64 48.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

   9. Simplified 48.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(-4 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(-4 \cdot U\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(-4 \cdot U\right), \left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot -4\right), \left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot n\right), Om\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right) \]

       9. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right) \]

   11. Applied egg-rr 58.1%

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot -4\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot -4\right) \cdot \frac{\ell \cdot \left(n \cdot \ell\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot \left(l\_m \cdot l\_m\right)\\ t_2 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot t\_1}{Om \cdot Om}\right)}\\ \mathbf{if}\;n \leq -3.05 \cdot 10^{+126}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{t\_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (* l_m l_m)))
        (t_2 (sqrt (* (* 2.0 (* n U)) (+ t (/ (* U* t_1) (* Om Om)))))))
   (if (<= n -3.05e+126)
     t_2
     (if (<= n 6.2e-67)
       (sqrt (* U (+ (* 2.0 (* n t)) (* -4.0 (/ t_1 Om)))))
       t_2))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double t_2 = sqrt(((2.0 * (n * U)) * (t + ((U_42_ * t_1) / (Om * Om)))));
	double tmp;
	if (n <= -3.05e+126) {
		tmp = t_2;
	} else if (n <= 6.2e-67) {
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = n * (l_m * l_m)
    t_2 = sqrt(((2.0d0 * (n * u)) * (t + ((u_42 * t_1) / (om * om)))))
    if (n <= (-3.05d+126)) then
        tmp = t_2
    else if (n <= 6.2d-67) then
        tmp = sqrt((u * ((2.0d0 * (n * t)) + ((-4.0d0) * (t_1 / om)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double t_2 = Math.sqrt(((2.0 * (n * U)) * (t + ((U_42_ * t_1) / (Om * Om)))));
	double tmp;
	if (n <= -3.05e+126) {
		tmp = t_2;
	} else if (n <= 6.2e-67) {
		tmp = Math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = n * (l_m * l_m)
	t_2 = math.sqrt(((2.0 * (n * U)) * (t + ((U_42_ * t_1) / (Om * Om)))))
	tmp = 0
	if n <= -3.05e+126:
		tmp = t_2
	elif n <= 6.2e-67:
		tmp = math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))))
	else:
		tmp = t_2
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * Float64(l_m * l_m))
	t_2 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(U_42_ * t_1) / Float64(Om * Om)))))
	tmp = 0.0
	if (n <= -3.05e+126)
		tmp = t_2;
	elseif (n <= 6.2e-67)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * Float64(n * t)) + Float64(-4.0 * Float64(t_1 / Om)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = n * (l_m * l_m);
	t_2 = sqrt(((2.0 * (n * U)) * (t + ((U_42_ * t_1) / (Om * Om)))));
	tmp = 0.0;
	if (n <= -3.05e+126)
		tmp = t_2;
	elseif (n <= 6.2e-67)
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(U$42$ * t$95$1), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3.05e+126], t$95$2, If[LessEqual[n, 6.2e-67], N[Sqrt[N[(U * N[(N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.05000000000000011e126 or 6.2000000000000005e-67 < n

   1. Initial program 47.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 64.7%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(n \cdot {\ell}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right) \]

   7. Simplified 56.2%

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

   if -3.05000000000000011e126 < n < 6.2000000000000005e-67

   1. Initial program 45.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 58.9%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om} + 2 \cdot \left(n \cdot t\right)\right)}, U\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right), U\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right), U\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot n\right), Om\right)\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right), Om\right)\right)\right), U\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

      9. *-lowering-*.f64 49.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

   9. Simplified 49.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)} \cdot U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.05 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{l\_m \cdot -2 + U* \cdot \left(l\_m \cdot \frac{n}{Om}\right)}{\frac{Om}{l\_m}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{\frac{Om \cdot Om}{n}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.05e+210)
   (sqrt
    (*
     U
     (*
      (* 2.0 n)
      (+ t (/ (+ (* l_m -2.0) (* U* (* l_m (/ n Om)))) (/ Om l_m))))))
   (* l_m (sqrt (* 2.0 (* (* n U) (+ (/ -2.0 Om) (/ U* (/ (* Om Om) n)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+210) {
		tmp = sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	} else {
		tmp = l_m * sqrt((2.0 * ((n * U) * ((-2.0 / Om) + (U_42_ / ((Om * Om) / n))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 2.05d+210) then
        tmp = sqrt((u * ((2.0d0 * n) * (t + (((l_m * (-2.0d0)) + (u_42 * (l_m * (n / om)))) / (om / l_m))))))
    else
        tmp = l_m * sqrt((2.0d0 * ((n * u) * (((-2.0d0) / om) + (u_42 / ((om * om) / n))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.05e+210) {
		tmp = Math.sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	} else {
		tmp = l_m * Math.sqrt((2.0 * ((n * U) * ((-2.0 / Om) + (U_42_ / ((Om * Om) / n))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.05e+210:
		tmp = math.sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))))
	else:
		tmp = l_m * math.sqrt((2.0 * ((n * U) * ((-2.0 / Om) + (U_42_ / ((Om * Om) / n))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.05e+210)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t + Float64(Float64(Float64(l_m * -2.0) + Float64(U_42_ * Float64(l_m * Float64(n / Om)))) / Float64(Om / l_m))))));
	else
		tmp = Float64(l_m * sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(Float64(-2.0 / Om) + Float64(U_42_ / Float64(Float64(Om * Om) / n)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.05e+210)
		tmp = sqrt((U * ((2.0 * n) * (t + (((l_m * -2.0) + (U_42_ * (l_m * (n / Om)))) / (Om / l_m))))));
	else
		tmp = l_m * sqrt((2.0 * ((n * U) * ((-2.0 / Om) + (U_42_ / ((Om * Om) / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.05e+210], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(U$42$ * N[(l$95$m * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 / Om), $MachinePrecision] + N[(U$42$ / N[(N[(Om * Om), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.05e210

   1. Initial program 48.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. /-lowering-/.f64 60.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 60.6%

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

   if 2.05e210 < l

   1. Initial program 16.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 38.5%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{U \cdot \color{blue}{\left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot n\right), \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \left(U* - U\right)\right), \left({Om}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U* - U\right)\right), \left({Om}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \left({Om}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \left(Om \cdot Om\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{Om}\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\mathsf{neg}\left(\frac{2}{Om}\right)\right)\right)\right)\right)\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{Om}\right)\right)\right)\right)\right) \]

   7. Simplified 79.2%

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

   8. Taylor expanded in U around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \color{blue}{\left(\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\left(\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot n\right), \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U* \cdot n}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(\frac{U* \cdot n}{{Om}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{n}{{Om}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{n}{{Om}^{2}}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \left({Om}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \left(Om \cdot Om\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \mathsf{*.f64}\left(Om, Om\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \mathsf{*.f64}\left(Om, Om\right)\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{Om}\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \mathsf{*.f64}\left(Om, Om\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{Om}\right)\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \mathsf{*.f64}\left(Om, Om\right)\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{Om}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \mathsf{*.f64}\left(Om, Om\right)\right)\right), \left(\frac{-2}{Om}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 85.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, \mathsf{*.f64}\left(Om, Om\right)\right)\right), \mathsf{/.f64}\left(-2, Om\right)\right)\right)\right)\right) \]

   10. Simplified 85.6%

       \[\leadsto \left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \left(U* \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2} \cdot \sqrt{\left(U \cdot n\right) \cdot \left(U* \cdot \frac{n}{Om \cdot Om} + \frac{-2}{Om}\right)}\right), \color{blue}{\ell}\right) \]

   12. Applied egg-rr 85.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{-2}{Om} + \frac{U*}{\frac{Om \cdot Om}{n}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot \left(l\_m \cdot l\_m\right)\\ \mathbf{if}\;l\_m \leq 9.6 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{-2 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;l\_m \leq 8.4 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{t\_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{U \cdot \left(U* \cdot t\_1\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (* l_m l_m))))
   (if (<= l_m 9.6e-83)
     (sqrt (* (* 2.0 n) (+ (* U t) (/ (* -2.0 (* U (* l_m l_m))) Om))))
     (if (<= l_m 8.4e+141)
       (sqrt (* U (+ (* 2.0 (* n t)) (* -4.0 (/ t_1 Om)))))
       (sqrt (* (* 2.0 n) (/ (* U (* U* t_1)) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double tmp;
	if (l_m <= 9.6e-83) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))));
	} else if (l_m <= 8.4e+141) {
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = sqrt(((2.0 * n) * ((U * (U_42_ * t_1)) / (Om * Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (l_m * l_m)
    if (l_m <= 9.6d-83) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + (((-2.0d0) * (u * (l_m * l_m))) / om))))
    else if (l_m <= 8.4d+141) then
        tmp = sqrt((u * ((2.0d0 * (n * t)) + ((-4.0d0) * (t_1 / om)))))
    else
        tmp = sqrt(((2.0d0 * n) * ((u * (u_42 * t_1)) / (om * om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double tmp;
	if (l_m <= 9.6e-83) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))));
	} else if (l_m <= 8.4e+141) {
		tmp = Math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * ((U * (U_42_ * t_1)) / (Om * Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = n * (l_m * l_m)
	tmp = 0
	if l_m <= 9.6e-83:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))))
	elif l_m <= 8.4e+141:
		tmp = math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))))
	else:
		tmp = math.sqrt(((2.0 * n) * ((U * (U_42_ * t_1)) / (Om * Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * Float64(l_m * l_m))
	tmp = 0.0
	if (l_m <= 9.6e-83)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(-2.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (l_m <= 8.4e+141)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * Float64(n * t)) + Float64(-4.0 * Float64(t_1 / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * Float64(U_42_ * t_1)) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = n * (l_m * l_m);
	tmp = 0.0;
	if (l_m <= 9.6e-83)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))));
	elseif (l_m <= 8.4e+141)
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	else
		tmp = sqrt(((2.0 * n) * ((U * (U_42_ * t_1)) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 9.6e-83], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(-2.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 8.4e+141], N[Sqrt[N[(U * N[(N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * N[(U$42$ * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.6000000000000003e-83

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om} + U \cdot t\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(U \cdot t\right), \left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \left(\frac{-2 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot {\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(U \cdot {\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      9. *-lowering-*.f64 46.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 46.3%

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

   if 9.6000000000000003e-83 < l < 8.3999999999999994e141

   1. Initial program 56.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om} + 2 \cdot \left(n \cdot t\right)\right)}, U\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right), U\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right), U\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot n\right), Om\right)\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right), Om\right)\right)\right), U\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

      9. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

   9. Simplified 47.7%

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

   if 8.3999999999999994e141 < l

   1. Initial program 14.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 15.0%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), \left(Om \cdot Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. *-lowering-*.f64 38.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 38.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.6 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{-2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot \left(l\_m \cdot l\_m\right)\\ \mathbf{if}\;l\_m \leq 6.6 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{-2 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;l\_m \leq 8.4 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{t\_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \frac{U* \cdot t\_1}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (* l_m l_m))))
   (if (<= l_m 6.6e-83)
     (sqrt (* (* 2.0 n) (+ (* U t) (/ (* -2.0 (* U (* l_m l_m))) Om))))
     (if (<= l_m 8.4e+141)
       (sqrt (* U (+ (* 2.0 (* n t)) (* -4.0 (/ t_1 Om)))))
       (sqrt (* U (* (* 2.0 n) (/ (* U* t_1) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double tmp;
	if (l_m <= 6.6e-83) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))));
	} else if (l_m <= 8.4e+141) {
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (l_m * l_m)
    if (l_m <= 6.6d-83) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + (((-2.0d0) * (u * (l_m * l_m))) / om))))
    else if (l_m <= 8.4d+141) then
        tmp = sqrt((u * ((2.0d0 * (n * t)) + ((-4.0d0) * (t_1 / om)))))
    else
        tmp = sqrt((u * ((2.0d0 * n) * ((u_42 * t_1) / (om * om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double tmp;
	if (l_m <= 6.6e-83) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))));
	} else if (l_m <= 8.4e+141) {
		tmp = Math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = Math.sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = n * (l_m * l_m)
	tmp = 0
	if l_m <= 6.6e-83:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))))
	elif l_m <= 8.4e+141:
		tmp = math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))))
	else:
		tmp = math.sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * Float64(l_m * l_m))
	tmp = 0.0
	if (l_m <= 6.6e-83)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(-2.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (l_m <= 8.4e+141)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * Float64(n * t)) + Float64(-4.0 * Float64(t_1 / Om)))));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(Float64(U_42_ * t_1) / Float64(Om * Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = n * (l_m * l_m);
	tmp = 0.0;
	if (l_m <= 6.6e-83)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((-2.0 * (U * (l_m * l_m))) / Om))));
	elseif (l_m <= 8.4e+141)
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	else
		tmp = sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 6.6e-83], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(-2.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 8.4e+141], N[Sqrt[N[(U * N[(N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U$42$ * t$95$1), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.5999999999999999e-83

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om} + U \cdot t\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot t + -2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(U \cdot t\right), \left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \left(\frac{-2 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot {\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(U \cdot {\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      9. *-lowering-*.f64 46.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 46.3%

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

   if 6.5999999999999999e-83 < l < 8.3999999999999994e141

   1. Initial program 56.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om} + 2 \cdot \left(n \cdot t\right)\right)}, U\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right), U\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right), U\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot n\right), Om\right)\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right), Om\right)\right)\right), U\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

      9. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

   9. Simplified 47.7%

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

   if 8.3999999999999994e141 < l

   1. Initial program 14.9%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}, \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(Om \cdot Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 38.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 38.0%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{-2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot \left(l\_m \cdot l\_m\right)\\ \mathbf{if}\;l\_m \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.55 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{t\_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \frac{U* \cdot t\_1}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (* l_m l_m))))
   (if (<= l_m 9e-83)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (* (* l_m l_m) -2.0) Om)))))
     (if (<= l_m 1.55e+141)
       (sqrt (* U (+ (* 2.0 (* n t)) (* -4.0 (/ t_1 Om)))))
       (sqrt (* U (* (* 2.0 n) (/ (* U* t_1) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double tmp;
	if (l_m <= 9e-83) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))));
	} else if (l_m <= 1.55e+141) {
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (l_m * l_m)
    if (l_m <= 9d-83) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l_m * l_m) * (-2.0d0)) / om)))))
    else if (l_m <= 1.55d+141) then
        tmp = sqrt((u * ((2.0d0 * (n * t)) + ((-4.0d0) * (t_1 / om)))))
    else
        tmp = sqrt((u * ((2.0d0 * n) * ((u_42 * t_1) / (om * om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (l_m * l_m);
	double tmp;
	if (l_m <= 9e-83) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))));
	} else if (l_m <= 1.55e+141) {
		tmp = Math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	} else {
		tmp = Math.sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = n * (l_m * l_m)
	tmp = 0
	if l_m <= 9e-83:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))))
	elif l_m <= 1.55e+141:
		tmp = math.sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))))
	else:
		tmp = math.sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * Float64(l_m * l_m))
	tmp = 0.0
	if (l_m <= 9e-83)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om)))));
	elseif (l_m <= 1.55e+141)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * Float64(n * t)) + Float64(-4.0 * Float64(t_1 / Om)))));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(Float64(U_42_ * t_1) / Float64(Om * Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = n * (l_m * l_m);
	tmp = 0.0;
	if (l_m <= 9e-83)
		tmp = sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))));
	elseif (l_m <= 1.55e+141)
		tmp = sqrt((U * ((2.0 * (n * t)) + (-4.0 * (t_1 / Om)))));
	else
		tmp = sqrt((U * ((2.0 * n) * ((U_42_ * t_1) / (Om * Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 9e-83], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.55e+141], N[Sqrt[N[(U * N[(N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U$42$ * t$95$1), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.99999999999999995e-83

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 46.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 46.4%

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

   if 8.99999999999999995e-83 < l < 1.55000000000000002e141

   1. Initial program 56.2%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om} + 2 \cdot \left(n \cdot t\right)\right)}, U\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right), U\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \left(\frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right), U\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot n\right), Om\right)\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right), Om\right)\right)\right), U\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

      9. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right), Om\right)\right)\right), U\right)\right) \]

   9. Simplified 47.7%

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

   if 1.55000000000000002e141 < l

   1. Initial program 14.9%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}, \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(Om \cdot Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 38.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 38.0%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot t\right) + -4 \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\\ \mathbf{if}\;l\_m \leq 6 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\_1\right)}\\ \mathbf{elif}\;l\_m \leq 3.8 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (+ t (/ (* (* l_m l_m) -2.0) Om))))
   (if (<= l_m 6e-79)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= l_m 3.8e+142)
       (sqrt (* U (* (* 2.0 n) t_1)))
       (sqrt (* U (* (* 2.0 n) (/ (* U* (* n (* l_m l_m))) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double tmp;
	if (l_m <= 6e-79) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (l_m <= 3.8e+142) {
		tmp = sqrt((U * ((2.0 * n) * t_1)));
	} else {
		tmp = sqrt((U * ((2.0 * n) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + (((l_m * l_m) * (-2.0d0)) / om)
    if (l_m <= 6d-79) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (l_m <= 3.8d+142) then
        tmp = sqrt((u * ((2.0d0 * n) * t_1)))
    else
        tmp = sqrt((u * ((2.0d0 * n) * ((u_42 * (n * (l_m * l_m))) / (om * om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double tmp;
	if (l_m <= 6e-79) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (l_m <= 3.8e+142) {
		tmp = Math.sqrt((U * ((2.0 * n) * t_1)));
	} else {
		tmp = Math.sqrt((U * ((2.0 * n) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t + (((l_m * l_m) * -2.0) / Om)
	tmp = 0
	if l_m <= 6e-79:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif l_m <= 3.8e+142:
		tmp = math.sqrt((U * ((2.0 * n) * t_1)))
	else:
		tmp = math.sqrt((U * ((2.0 * n) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om))
	tmp = 0.0
	if (l_m <= 6e-79)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (l_m <= 3.8e+142)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * t_1)));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / Float64(Om * Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = t + (((l_m * l_m) * -2.0) / Om);
	tmp = 0.0;
	if (l_m <= 6e-79)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (l_m <= 3.8e+142)
		tmp = sqrt((U * ((2.0 * n) * t_1)));
	else
		tmp = sqrt((U * ((2.0 * n) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 6e-79], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.8e+142], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.99999999999999999e-79

   1. Initial program 48.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 45.1%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 46.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 46.2%

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

   if 5.99999999999999999e-79 < l < 3.7999999999999999e142

   1. Initial program 57.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 67.5%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}, \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      6. *-lowering-*.f64 46.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 46.6%

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

   if 3.7999999999999999e142 < l

   1. Initial program 14.9%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}, \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(Om \cdot Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      7. *-lowering-*.f64 38.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 38.0%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\\ \mathbf{if}\;l\_m \leq 1.6 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\_1\right)}\\ \mathbf{elif}\;l\_m \leq 3.8 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (+ t (/ (* (* l_m l_m) -2.0) Om))))
   (if (<= l_m 1.6e-78)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= l_m 3.8e+142)
       (sqrt (* U (* (* 2.0 n) t_1)))
       (sqrt (* (* (* 2.0 n) U) (/ (* U* (* n (* l_m l_m))) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double tmp;
	if (l_m <= 1.6e-78) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (l_m <= 3.8e+142) {
		tmp = sqrt((U * ((2.0 * n) * t_1)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + (((l_m * l_m) * (-2.0d0)) / om)
    if (l_m <= 1.6d-78) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (l_m <= 3.8d+142) then
        tmp = sqrt((u * ((2.0d0 * n) * t_1)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * ((u_42 * (n * (l_m * l_m))) / (om * om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double tmp;
	if (l_m <= 1.6e-78) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (l_m <= 3.8e+142) {
		tmp = Math.sqrt((U * ((2.0 * n) * t_1)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t + (((l_m * l_m) * -2.0) / Om)
	tmp = 0
	if l_m <= 1.6e-78:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif l_m <= 3.8e+142:
		tmp = math.sqrt((U * ((2.0 * n) * t_1)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om))
	tmp = 0.0
	if (l_m <= 1.6e-78)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (l_m <= 3.8e+142)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * t_1)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = t + (((l_m * l_m) * -2.0) / Om);
	tmp = 0.0;
	if (l_m <= 1.6e-78)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (l_m <= 3.8e+142)
		tmp = sqrt((U * ((2.0 * n) * t_1)));
	else
		tmp = sqrt((((2.0 * n) * U) * ((U_42_ * (n * (l_m * l_m))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 1.6e-78], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 3.8e+142], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.6e-78

   1. Initial program 48.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 45.1%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 46.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 46.2%

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

   if 1.6e-78 < l < 3.7999999999999999e142

   1. Initial program 57.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 67.5%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}, \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      6. *-lowering-*.f64 46.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 46.6%

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

   if 3.7999999999999999e142 < l

   1. Initial program 14.9%

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

   3. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(n \cdot {\ell}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right) \]

      8. *-lowering-*.f64 37.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right) \]

   5. Simplified 37.9%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\\ t_2 := U \cdot \left(n \cdot t\right)\\ \mathbf{if}\;l\_m \leq 3.3 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\_1\right)}\\ \mathbf{elif}\;l\_m \leq 6 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(t\_2 \cdot t\_2\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (+ t (/ (* (* l_m l_m) -2.0) Om))) (t_2 (* U (* n t))))
   (if (<= l_m 3.3e-83)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= l_m 6e+150)
       (sqrt (* U (* (* 2.0 n) t_1)))
       (pow (* 4.0 (* t_2 t_2)) 0.25)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double t_2 = U * (n * t);
	double tmp;
	if (l_m <= 3.3e-83) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (l_m <= 6e+150) {
		tmp = sqrt((U * ((2.0 * n) * t_1)));
	} else {
		tmp = pow((4.0 * (t_2 * t_2)), 0.25);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t + (((l_m * l_m) * (-2.0d0)) / om)
    t_2 = u * (n * t)
    if (l_m <= 3.3d-83) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (l_m <= 6d+150) then
        tmp = sqrt((u * ((2.0d0 * n) * t_1)))
    else
        tmp = (4.0d0 * (t_2 * t_2)) ** 0.25d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (((l_m * l_m) * -2.0) / Om);
	double t_2 = U * (n * t);
	double tmp;
	if (l_m <= 3.3e-83) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (l_m <= 6e+150) {
		tmp = Math.sqrt((U * ((2.0 * n) * t_1)));
	} else {
		tmp = Math.pow((4.0 * (t_2 * t_2)), 0.25);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t + (((l_m * l_m) * -2.0) / Om)
	t_2 = U * (n * t)
	tmp = 0
	if l_m <= 3.3e-83:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif l_m <= 6e+150:
		tmp = math.sqrt((U * ((2.0 * n) * t_1)))
	else:
		tmp = math.pow((4.0 * (t_2 * t_2)), 0.25)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om))
	t_2 = Float64(U * Float64(n * t))
	tmp = 0.0
	if (l_m <= 3.3e-83)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (l_m <= 6e+150)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * t_1)));
	else
		tmp = Float64(4.0 * Float64(t_2 * t_2)) ^ 0.25;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = t + (((l_m * l_m) * -2.0) / Om);
	t_2 = U * (n * t);
	tmp = 0.0;
	if (l_m <= 3.3e-83)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (l_m <= 6e+150)
		tmp = sqrt((U * ((2.0 * n) * t_1)));
	else
		tmp = (4.0 * (t_2 * t_2)) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 3.3e-83], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 6e+150], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(4.0 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.2999999999999999e-83

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 46.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 46.4%

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

   if 3.2999999999999999e-83 < l < 6.00000000000000025e150

   1. Initial program 55.0%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}, \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

      6. *-lowering-*.f64 45.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right), U\right)\right) \]

   9. Simplified 45.2%

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

   if 6.00000000000000025e150 < l

   1. Initial program 12.7%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 50.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-lowering-*.f64 12.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

   7. Simplified 12.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot n\right), \left(U \cdot t\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 2\right), \left(U \cdot t\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \left(U \cdot t\right)\right)\right) \]

      8. *-lowering-*.f64 9.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \mathsf{*.f64}\left(U, t\right)\right)\right) \]

   9. Applied egg-rr 9.0%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. *-commutative N/A

          \[\leadsto {\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{\frac{1}{2}} \]

       3. *-commutative N/A

          \[\leadsto {\left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right)}^{\frac{1}{2}} \]

       4. associate-*l* N/A

          \[\leadsto {\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{\frac{1}{2}} \]

       5. *-commutative N/A

          \[\leadsto {\left(t \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}} \]

       6. associate-*l* N/A

          \[\leadsto {\left(t \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\left(t \cdot \left(\left(2 \cdot U\right) \cdot n\right)\right)}^{\frac{1}{2}} \]

       8. *-commutative N/A

          \[\leadsto {\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}^{\frac{1}{2}} \]

       9. associate-*r* N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\frac{1}{2}} \]

       10. metadata-eval N/A

           \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

       11. metadata-eval N/A

           \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)} \]

       12. metadata-eval N/A

           \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{\color{blue}{2}}\right)} \]

       13. pow-prod-up N/A

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

       14. pow-prod-down N/A

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

       15. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 25.6%

       \[\leadsto \color{blue}{{\left(4 \cdot \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)}^{0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(\left(U \cdot \left(n \cdot t\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;U \leq -7.2 \cdot 10^{-14}:\\ \;\;\;\;{\left(t \cdot t\_1\right)}^{0.5}\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(l\_m \cdot l\_m\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= U -7.2e-14)
     (pow (* t t_1) 0.5)
     (if (<= U 1.2e+32)
       (sqrt (* (* 2.0 n) (* U (+ t (/ (* (* l_m l_m) -2.0) Om)))))
       (sqrt (* t_1 (- t (/ (* 2.0 (* l_m l_m)) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (U <= -7.2e-14) {
		tmp = pow((t * t_1), 0.5);
	} else if (U <= 1.2e+32) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))));
	} else {
		tmp = sqrt((t_1 * (t - ((2.0 * (l_m * l_m)) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (u <= (-7.2d-14)) then
        tmp = (t * t_1) ** 0.5d0
    else if (u <= 1.2d+32) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l_m * l_m) * (-2.0d0)) / om)))))
    else
        tmp = sqrt((t_1 * (t - ((2.0d0 * (l_m * l_m)) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (U <= -7.2e-14) {
		tmp = Math.pow((t * t_1), 0.5);
	} else if (U <= 1.2e+32) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))));
	} else {
		tmp = Math.sqrt((t_1 * (t - ((2.0 * (l_m * l_m)) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if U <= -7.2e-14:
		tmp = math.pow((t * t_1), 0.5)
	elif U <= 1.2e+32:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))))
	else:
		tmp = math.sqrt((t_1 * (t - ((2.0 * (l_m * l_m)) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (U <= -7.2e-14)
		tmp = Float64(t * t_1) ^ 0.5;
	elseif (U <= 1.2e+32)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l_m * l_m) * -2.0) / Om)))));
	else
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (U <= -7.2e-14)
		tmp = (t * t_1) ^ 0.5;
	elseif (U <= 1.2e+32)
		tmp = sqrt(((2.0 * n) * (U * (t + (((l_m * l_m) * -2.0) / Om)))));
	else
		tmp = sqrt((t_1 * (t - ((2.0 * (l_m * l_m)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -7.2e-14], N[Power[N[(t * t$95$1), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 1.2e+32], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if U < -7.1999999999999996e-14

   1. Initial program 47.3%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 62.6%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

   7. Simplified 41.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right), \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right), \frac{1}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(t \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right), \frac{1}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(2 \cdot n\right) \cdot U\right)\right), \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(n \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \left(U \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 48.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{{\left(t \cdot \left(2 \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

   if -7.1999999999999996e-14 < U < 1.19999999999999996e32

   1. Initial program 42.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   4. Applied egg-rr 46.2%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(-2 \cdot {\ell}^{2}\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2}\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 43.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. Simplified 43.6%

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

   if 1.19999999999999996e32 < U

   1. Initial program 60.0%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 74.5%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \left(\frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right) \]

      8. *-lowering-*.f64 54.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 54.7%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -7.2 \cdot 10^{-14}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{n \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (* t (* n (* 2.0 U))) 0.5)))
   (if (<= t -8.2e-134)
     t_1
     (if (<= t 1.08e-75) (sqrt (* -4.0 (* U (/ (* n (* l_m l_m)) Om)))) t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow((t * (n * (2.0 * U))), 0.5);
	double tmp;
	if (t <= -8.2e-134) {
		tmp = t_1;
	} else if (t <= 1.08e-75) {
		tmp = sqrt((-4.0 * (U * ((n * (l_m * l_m)) / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * (n * (2.0d0 * u))) ** 0.5d0
    if (t <= (-8.2d-134)) then
        tmp = t_1
    else if (t <= 1.08d-75) then
        tmp = sqrt(((-4.0d0) * (u * ((n * (l_m * l_m)) / om))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.pow((t * (n * (2.0 * U))), 0.5);
	double tmp;
	if (t <= -8.2e-134) {
		tmp = t_1;
	} else if (t <= 1.08e-75) {
		tmp = Math.sqrt((-4.0 * (U * ((n * (l_m * l_m)) / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.pow((t * (n * (2.0 * U))), 0.5)
	tmp = 0
	if t <= -8.2e-134:
		tmp = t_1
	elif t <= 1.08e-75:
		tmp = math.sqrt((-4.0 * (U * ((n * (l_m * l_m)) / Om))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t * Float64(n * Float64(2.0 * U))) ^ 0.5
	tmp = 0.0
	if (t <= -8.2e-134)
		tmp = t_1;
	elseif (t <= 1.08e-75)
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64(Float64(n * Float64(l_m * l_m)) / Om))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (t * (n * (2.0 * U))) ^ 0.5;
	tmp = 0.0;
	if (t <= -8.2e-134)
		tmp = t_1;
	elseif (t <= 1.08e-75)
		tmp = sqrt((-4.0 * (U * ((n * (l_m * l_m)) / Om))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[t, -8.2e-134], t$95$1, If[LessEqual[t, 1.08e-75], N[Sqrt[N[(-4.0 * N[(U * N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.2000000000000004e-134 or 1.08e-75 < t

   1. Initial program 51.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.4%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-lowering-*.f64 45.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

   7. Simplified 45.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot n\right), \left(U \cdot t\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 2\right), \left(U \cdot t\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \left(U \cdot t\right)\right)\right) \]

      8. *-lowering-*.f64 45.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \mathsf{*.f64}\left(U, t\right)\right)\right) \]

   9. Applied egg-rr 45.3%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. associate-*r* N/A

          \[\leadsto {\left(\left(\left(n \cdot 2\right) \cdot U\right) \cdot t\right)}^{\frac{1}{2}} \]

       3. unpow-prod-down N/A

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

       4. associate-*l* N/A

          \[\leadsto {\left(n \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}} \cdot {t}^{\frac{1}{2}} \]

       5. *-commutative N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\frac{1}{2}} \cdot {t}^{\frac{1}{2}} \]

       6. unpow-prod-down N/A

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

       7. associate-*r* N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\frac{1}{2}} \]

       8. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right), \frac{1}{2}\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(t \cdot \left(\left(2 \cdot U\right) \cdot n\right)\right), \frac{1}{2}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(2 \cdot U\right) \cdot n\right)\right), \frac{1}{2}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(n \cdot \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

       14. *-lowering-*.f64 49.5%

           \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right), \frac{1}{2}\right) \]

   11. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{{\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}} \]

   if -8.2000000000000004e-134 < t < 1.08e-75

   1. Initial program 38.0%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 50.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot n\right)\right), U\right)\right) \]

   6. Applied egg-rr 49.6%

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

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

      13. *-lowering-*.f64 29.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right)\right) \]

   9. Simplified 29.8%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left(\frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot n\right), Om\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\left(n \cdot {\ell}^{2}\right), Om\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right) \]

       8. *-lowering-*.f64 26.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right) \]

   12. Simplified 26.2%

       \[\leadsto \sqrt{\color{blue}{-4 \cdot \left(U \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot t\right)\\ \mathbf{if}\;l\_m \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(t\_1 \cdot t\_1\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n t))))
   (if (<= l_m 1.45e+151)
     (sqrt (* (* 2.0 (* n U)) (- t (/ (* 2.0 (* l_m l_m)) Om))))
     (pow (* 4.0 (* t_1 t_1)) 0.25))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * t);
	double tmp;
	if (l_m <= 1.45e+151) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l_m * l_m)) / Om))));
	} else {
		tmp = pow((4.0 * (t_1 * t_1)), 0.25);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (n * t)
    if (l_m <= 1.45d+151) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((2.0d0 * (l_m * l_m)) / om))))
    else
        tmp = (4.0d0 * (t_1 * t_1)) ** 0.25d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * t);
	double tmp;
	if (l_m <= 1.45e+151) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l_m * l_m)) / Om))));
	} else {
		tmp = Math.pow((4.0 * (t_1 * t_1)), 0.25);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n * t)
	tmp = 0
	if l_m <= 1.45e+151:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l_m * l_m)) / Om))))
	else:
		tmp = math.pow((4.0 * (t_1 * t_1)), 0.25)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * t))
	tmp = 0.0
	if (l_m <= 1.45e+151)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(2.0 * Float64(l_m * l_m)) / Om))));
	else
		tmp = Float64(4.0 * Float64(t_1 * t_1)) ^ 0.25;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n * t);
	tmp = 0.0;
	if (l_m <= 1.45e+151)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l_m * l_m)) / Om))));
	else
		tmp = (4.0 * (t_1 * t_1)) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 1.45e+151], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.45000000000000009e151

   1. Initial program 50.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 57.7%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \left(\frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), Om\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), Om\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), Om\right)\right)\right)\right) \]

      8. *-lowering-*.f64 44.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{\_.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right)\right)\right)\right) \]

   7. Simplified 44.5%

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

   if 1.45000000000000009e151 < l

   1. Initial program 12.7%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 50.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-lowering-*.f64 12.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

   7. Simplified 12.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot n\right), \left(U \cdot t\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 2\right), \left(U \cdot t\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \left(U \cdot t\right)\right)\right) \]

      8. *-lowering-*.f64 9.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \mathsf{*.f64}\left(U, t\right)\right)\right) \]

   9. Applied egg-rr 9.0%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

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

       2. *-commutative N/A

          \[\leadsto {\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{\frac{1}{2}} \]

       3. *-commutative N/A

          \[\leadsto {\left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right)}^{\frac{1}{2}} \]

       4. associate-*l* N/A

          \[\leadsto {\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{\frac{1}{2}} \]

       5. *-commutative N/A

          \[\leadsto {\left(t \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}} \]

       6. associate-*l* N/A

          \[\leadsto {\left(t \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}} \]

       7. *-commutative N/A

          \[\leadsto {\left(t \cdot \left(\left(2 \cdot U\right) \cdot n\right)\right)}^{\frac{1}{2}} \]

       8. *-commutative N/A

          \[\leadsto {\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)}^{\frac{1}{2}} \]

       9. associate-*r* N/A

          \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\frac{1}{2}} \]

       10. metadata-eval N/A

           \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

       11. metadata-eval N/A

           \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)} \]

       12. metadata-eval N/A

           \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{\color{blue}{2}}\right)} \]

       13. pow-prod-up N/A

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

       14. pow-prod-down N/A

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

       15. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 25.6%

       \[\leadsto \color{blue}{{\left(4 \cdot \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(4 \cdot \left(\left(U \cdot \left(n \cdot t\right)\right) \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 2e+143) (sqrt (* (* 2.0 n) (* U t))) (sqrt (* 2.0 (* t (* n U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 2e+143) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 2d+143) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 2e+143) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= 2e+143:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 2e+143)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= 2e+143)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, 2e+143], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 2e143

   1. Initial program 42.0%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 52.7%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-lowering-*.f64 27.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

   7. Simplified 27.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot n\right), \left(U \cdot t\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 2\right), \left(U \cdot t\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \left(U \cdot t\right)\right)\right) \]

      8. *-lowering-*.f64 30.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \mathsf{*.f64}\left(U, t\right)\right)\right) \]

   9. Applied egg-rr 30.2%

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

   if 2e143 < Om

   1. Initial program 76.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 85.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

   7. Simplified 71.2%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ {\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* t (* n (* 2.0 U))) 0.5))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return pow((t * (n * (2.0 * U))), 0.5);
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (t * (n * (2.0d0 * u))) ** 0.5d0
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.pow((t * (n * (2.0 * U))), 0.5);
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.pow((t * (n * (2.0 * U))), 0.5)

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return Float64(t * Float64(n * Float64(2.0 * U))) ^ 0.5
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = (t * (n * (2.0 * U))) ^ 0.5;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Power[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 46.4%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

3. Simplified 56.9%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

   4. *-lowering-*.f64 33.5%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

7. Simplified 33.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot n\right), \left(U \cdot t\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(n \cdot 2\right), \left(U \cdot t\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \left(U \cdot t\right)\right)\right) \]

   8. *-lowering-*.f64 33.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), \mathsf{*.f64}\left(U, t\right)\right)\right) \]

9. Applied egg-rr 33.6%

   \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}} \]
10. Step-by-step derivation

    1. pow1/2 N/A

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

    2. associate-*r* N/A

       \[\leadsto {\left(\left(\left(n \cdot 2\right) \cdot U\right) \cdot t\right)}^{\frac{1}{2}} \]

    3. unpow-prod-down N/A

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

    4. associate-*l* N/A

       \[\leadsto {\left(n \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}} \cdot {t}^{\frac{1}{2}} \]

    5. *-commutative N/A

       \[\leadsto {\left(\left(2 \cdot U\right) \cdot n\right)}^{\frac{1}{2}} \cdot {t}^{\frac{1}{2}} \]

    6. unpow-prod-down N/A

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

    7. associate-*r* N/A

       \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\frac{1}{2}} \]

    8. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

    9. associate-*r* N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right), \frac{1}{2}\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{pow.f64}\left(\left(t \cdot \left(\left(2 \cdot U\right) \cdot n\right)\right), \frac{1}{2}\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(2 \cdot U\right) \cdot n\right)\right), \frac{1}{2}\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(n \cdot \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

    14. *-lowering-*.f64 36.3%

        \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right), \frac{1}{2}\right) \]

11. Applied egg-rr 36.3%

    \[\leadsto \color{blue}{{\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}} \]
12. Add Preprocessing

Alternative 21: 38.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ {\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* t (* 2.0 (* n U))) 0.5))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return pow((t * (2.0 * (n * U))), 0.5);
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (t * (2.0d0 * (n * u))) ** 0.5d0
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.pow((t * (2.0 * (n * U))), 0.5);
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.pow((t * (2.0 * (n * U))), 0.5)

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = (t * (2.0 * (n * U))) ^ 0.5;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 46.4%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

3. Simplified 56.9%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

   4. *-lowering-*.f64 33.5%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

7. Simplified 33.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. pow1/2 N/A

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

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right), \frac{1}{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right), \frac{1}{2}\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \frac{1}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(t \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right), \frac{1}{2}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(2 \cdot n\right) \cdot U\right)\right), \frac{1}{2}\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(n \cdot U\right)\right)\right), \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n\right)\right)\right), \frac{1}{2}\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \left(U \cdot n\right)\right)\right), \frac{1}{2}\right) \]

   11. *-lowering-*.f64 36.2%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

9. Applied egg-rr 36.2%

   \[\leadsto \color{blue}{{\left(t \cdot \left(2 \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

10. Final simplification 36.2%

    \[\leadsto {\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5} \]
11. Add Preprocessing

Alternative 22: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 2.0 (* t (* n U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((2.0 * (t * (n * U))));
}

Fortran

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((2.0 * (t * (n * U))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((2.0 * (t * (n * U))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(t * Float64(n * U))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((2.0 * (t * (n * U))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 46.4%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

3. Simplified 56.9%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

   4. *-lowering-*.f64 33.5%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), t\right)\right)\right) \]

7. Simplified 33.5%

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

8. Final simplification 33.5%

   \[\leadsto \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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*))))))