Toniolo and Linder, Equation (13)

Percentage Accurate: 49.4% → 69.9%

Time: 25.4s

Alternatives: 24

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: 49.4% 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: 69.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{l\_m} \cdot {\left(2 \cdot \left(U \cdot \frac{n \cdot t}{l\_m}\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_1 \leq 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{l\_m} \cdot {\left(\frac{-2 \cdot \left(U \cdot \left(\left(n \cdot l\_m\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

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 (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (* (sqrt l_m) (pow (* 2.0 (* U (/ (* n t) l_m))) 0.5))
     (if (<= t_1 1e+144)
       t_1
       (*
        (sqrt l_m)
        (pow
         (/ (* -2.0 (* U (* (* n l_m) (+ 2.0 (/ (* n (- U U*)) Om))))) Om)
         0.5))))))

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 - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(l_m) * pow((2.0 * (U * ((n * t) / l_m))), 0.5);
	} else if (t_1 <= 1e+144) {
		tmp = t_1;
	} else {
		tmp = sqrt(l_m) * pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5);
	}
	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 - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(l_m) * ((2.0d0 * (u * ((n * t) / l_m))) ** 0.5d0)
    else if (t_1 <= 1d+144) then
        tmp = t_1
    else
        tmp = sqrt(l_m) * ((((-2.0d0) * (u * ((n * l_m) * (2.0d0 + ((n * (u - u_42)) / om))))) / om) ** 0.5d0)
    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 - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(l_m) * Math.pow((2.0 * (U * ((n * t) / l_m))), 0.5);
	} else if (t_1 <= 1e+144) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(l_m) * Math.pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5);
	}
	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 - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(l_m) * math.pow((2.0 * (U * ((n * t) / l_m))), 0.5)
	elif t_1 <= 1e+144:
		tmp = t_1
	else:
		tmp = math.sqrt(l_m) * math.pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(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_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(l_m) * (Float64(2.0 * Float64(U * Float64(Float64(n * t) / l_m))) ^ 0.5));
	elseif (t_1 <= 1e+144)
		tmp = t_1;
	else
		tmp = Float64(sqrt(l_m) * (Float64(Float64(-2.0 * Float64(U * Float64(Float64(n * l_m) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))))) / Om) ^ 0.5));
	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 - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(l_m) * ((2.0 * (U * ((n * t) / l_m))) ^ 0.5);
	elseif (t_1 <= 1e+144)
		tmp = t_1;
	else
		tmp = sqrt(l_m) * (((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om) ^ 0.5);
	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[(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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[l$95$m], $MachinePrecision] * N[Power[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+144], t$95$1, N[(N[Sqrt[l$95$m], $MachinePrecision] * N[Power[N[(N[(-2.0 * N[(U * N[(N[(n * l$95$m), $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.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 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in l around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   7. Simplified 9.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

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

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

   9. Applied egg-rr 7.9%

      \[\leadsto \color{blue}{\sqrt{\ell} \cdot {\left(\left(\ell \cdot \left(\frac{t \cdot \left(U \cdot n\right)}{\ell \cdot \ell} - \frac{U \cdot \left(n \cdot \left(2 - \frac{\left(U* - U\right) \cdot n}{Om}\right)\right)}{Om}\right)\right) \cdot 2\right)}^{0.5}} \]

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{U \cdot \left(n \cdot t\right)}{\ell}\right)}, 2\right), \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

       1. associate-/l* N/A

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

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

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

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

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

       4. *-lowering-*.f64 21.7%

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

   12. Simplified 21.7%

       \[\leadsto \sqrt{\ell} \cdot {\left(\color{blue}{\left(U \cdot \frac{n \cdot t}{\ell}\right)} \cdot 2\right)}^{0.5} \]

   if 0.0 < (sqrt.f64 (*.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*))))) < 1.00000000000000002e144

   1. Initial program 98.6%

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

   if 1.00000000000000002e144 < (sqrt.f64 (*.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 24.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in l around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   7. Simplified 25.6%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

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

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

   9. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{\sqrt{\ell} \cdot {\left(\left(\ell \cdot \left(\frac{t \cdot \left(U \cdot n\right)}{\ell \cdot \ell} - \frac{U \cdot \left(n \cdot \left(2 - \frac{\left(U* - U\right) \cdot n}{Om}\right)\right)}{Om}\right)\right) \cdot 2\right)}^{0.5}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

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

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

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

       11. --lowering--.f64 25.4%

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

   12. Simplified 25.4%

       \[\leadsto \sqrt{\ell} \cdot {\color{blue}{\left(\frac{-2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}}^{0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\ell} \cdot {\left(2 \cdot \left(U \cdot \frac{n \cdot t}{\ell}\right)\right)}^{0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+144}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell} \cdot {\left(\frac{-2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(t + \frac{\frac{U*}{\frac{\frac{Om}{l\_m}}{n}} + l\_m \cdot -2}{\frac{Om}{l\_m}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;l\_m \leq 7.2 \cdot 10^{+186}:\\ \;\;\;\;\sqrt{U \cdot \left(\frac{n \cdot t}{l\_m \cdot l\_m} + \frac{n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{l\_m} \cdot {\left(\frac{-2 \cdot \left(U \cdot \left(\left(n \cdot l\_m\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.5e-56)
   (sqrt
    (*
     (+ t (/ (+ (/ U* (/ (/ Om l_m) n)) (* l_m -2.0)) (/ Om l_m)))
     (* 2.0 (* n U))))
   (if (<= l_m 7.2e+186)
     (*
      (sqrt
       (* U (+ (/ (* n t) (* l_m l_m)) (/ (* n (- (/ (* n U*) Om) 2.0)) Om))))
      (* l_m (sqrt 2.0)))
     (*
      (sqrt l_m)
      (pow
       (/ (* -2.0 (* U (* (* n l_m) (+ 2.0 (/ (* n (- U U*)) Om))))) Om)
       0.5)))))

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 <= 2.5e-56) {
		tmp = sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))));
	} else if (l_m <= 7.2e+186) {
		tmp = sqrt((U * (((n * t) / (l_m * l_m)) + ((n * (((n * U_42_) / Om) - 2.0)) / Om)))) * (l_m * sqrt(2.0));
	} else {
		tmp = sqrt(l_m) * pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5);
	}
	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 <= 2.5d-56) then
        tmp = sqrt(((t + (((u_42 / ((om / l_m) / n)) + (l_m * (-2.0d0))) / (om / l_m))) * (2.0d0 * (n * u))))
    else if (l_m <= 7.2d+186) then
        tmp = sqrt((u * (((n * t) / (l_m * l_m)) + ((n * (((n * u_42) / om) - 2.0d0)) / om)))) * (l_m * sqrt(2.0d0))
    else
        tmp = sqrt(l_m) * ((((-2.0d0) * (u * ((n * l_m) * (2.0d0 + ((n * (u - u_42)) / om))))) / om) ** 0.5d0)
    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 <= 2.5e-56) {
		tmp = Math.sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))));
	} else if (l_m <= 7.2e+186) {
		tmp = Math.sqrt((U * (((n * t) / (l_m * l_m)) + ((n * (((n * U_42_) / Om) - 2.0)) / Om)))) * (l_m * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(l_m) * Math.pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.5e-56:
		tmp = math.sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))))
	elif l_m <= 7.2e+186:
		tmp = math.sqrt((U * (((n * t) / (l_m * l_m)) + ((n * (((n * U_42_) / Om) - 2.0)) / Om)))) * (l_m * math.sqrt(2.0))
	else:
		tmp = math.sqrt(l_m) * math.pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.5e-56)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(U_42_ / Float64(Float64(Om / l_m) / n)) + Float64(l_m * -2.0)) / Float64(Om / l_m))) * Float64(2.0 * Float64(n * U))));
	elseif (l_m <= 7.2e+186)
		tmp = Float64(sqrt(Float64(U * Float64(Float64(Float64(n * t) / Float64(l_m * l_m)) + Float64(Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0)) / Om)))) * Float64(l_m * sqrt(2.0)));
	else
		tmp = Float64(sqrt(l_m) * (Float64(Float64(-2.0 * Float64(U * Float64(Float64(n * l_m) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))))) / Om) ^ 0.5));
	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 <= 2.5e-56)
		tmp = sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))));
	elseif (l_m <= 7.2e+186)
		tmp = sqrt((U * (((n * t) / (l_m * l_m)) + ((n * (((n * U_42_) / Om) - 2.0)) / Om)))) * (l_m * sqrt(2.0));
	else
		tmp = sqrt(l_m) * (((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om) ^ 0.5);
	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, 2.5e-56], N[Sqrt[N[(N[(t + N[(N[(N[(U$42$ / N[(N[(Om / l$95$m), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 7.2e+186], N[(N[Sqrt[N[(U * N[(N[(N[(n * t), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[l$95$m], $MachinePrecision] * N[Power[N[(N[(-2.0 * N[(U * N[(N[(n * l$95$m), $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.49999999999999999e-56

   1. Initial program 56.3%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 59.6%

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

   7. Simplified 59.6%

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

      1. *-commutative N/A

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

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

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

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

      7. /-lowering-/.f64 60.7%

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

   9. Applied egg-rr 60.7%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

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

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

   11. Applied egg-rr 60.8%

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

   if 2.49999999999999999e-56 < l < 7.2000000000000003e186

   1. Initial program 37.0%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in l around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   7. Simplified 48.7%

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

   8. Taylor expanded in U around 0

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

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

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

   10. Simplified 60.8%

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

   if 7.2000000000000003e186 < l

   1. Initial program 13.5%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in l around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   7. Simplified 24.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

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

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

   9. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\sqrt{\ell} \cdot {\left(\left(\ell \cdot \left(\frac{t \cdot \left(U \cdot n\right)}{\ell \cdot \ell} - \frac{U \cdot \left(n \cdot \left(2 - \frac{\left(U* - U\right) \cdot n}{Om}\right)\right)}{Om}\right)\right) \cdot 2\right)}^{0.5}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

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

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

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

       11. --lowering--.f64 83.1%

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

   12. Simplified 83.1%

       \[\leadsto \sqrt{\ell} \cdot {\color{blue}{\left(\frac{-2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}}^{0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+186}:\\ \;\;\;\;\sqrt{U \cdot \left(\frac{n \cdot t}{\ell \cdot \ell} + \frac{n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell} \cdot {\left(\frac{-2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(\frac{U*}{\frac{\frac{Om}{l\_m}}{n}} + l\_m \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{l\_m} \cdot {\left(\frac{-2 \cdot \left(U \cdot \left(\left(n \cdot l\_m\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.65e+66)
   (sqrt
    (*
     2.0
     (*
      (* n U)
      (+ t (* (/ l_m Om) (+ (/ U* (/ (/ Om l_m) n)) (* l_m -2.0)))))))
   (*
    (sqrt l_m)
    (pow
     (/ (* -2.0 (* U (* (* n l_m) (+ 2.0 (/ (* n (- U U*)) Om))))) Om)
     0.5))))

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 <= 1.65e+66) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	} else {
		tmp = sqrt(l_m) * pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5);
	}
	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 <= 1.65d+66) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m / om) * ((u_42 / ((om / l_m) / n)) + (l_m * (-2.0d0))))))))
    else
        tmp = sqrt(l_m) * ((((-2.0d0) * (u * ((n * l_m) * (2.0d0 + ((n * (u - u_42)) / om))))) / om) ** 0.5d0)
    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 <= 1.65e+66) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	} else {
		tmp = Math.sqrt(l_m) * Math.pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.65e+66:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))))
	else:
		tmp = math.sqrt(l_m) * math.pow(((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.65e+66)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m / Om) * Float64(Float64(U_42_ / Float64(Float64(Om / l_m) / n)) + Float64(l_m * -2.0)))))));
	else
		tmp = Float64(sqrt(l_m) * (Float64(Float64(-2.0 * Float64(U * Float64(Float64(n * l_m) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))))) / Om) ^ 0.5));
	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 <= 1.65e+66)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	else
		tmp = sqrt(l_m) * (((-2.0 * (U * ((n * l_m) * (2.0 + ((n * (U - U_42_)) / Om))))) / Om) ^ 0.5);
	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, 1.65e+66], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ / N[(N[(Om / l$95$m), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[l$95$m], $MachinePrecision] * N[Power[N[(N[(-2.0 * N[(U * N[(N[(n * l$95$m), $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6500000000000001e66

   1. Initial program 56.0%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 58.9%

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

   7. Simplified 58.9%

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

      1. *-commutative N/A

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

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

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

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

      7. /-lowering-/.f64 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

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

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

       6. associate-/r* N/A

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

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

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

       8. /-lowering-/.f64 60.4%

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

   11. Applied egg-rr 60.4%

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

   if 1.6500000000000001e66 < l

   1. Initial program 19.5%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in l around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

   7. Simplified 32.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

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

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

   9. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{\sqrt{\ell} \cdot {\left(\left(\ell \cdot \left(\frac{t \cdot \left(U \cdot n\right)}{\ell \cdot \ell} - \frac{U \cdot \left(n \cdot \left(2 - \frac{\left(U* - U\right) \cdot n}{Om}\right)\right)}{Om}\right)\right) \cdot 2\right)}^{0.5}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\ell\right), \mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

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

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

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

       11. --lowering--.f64 66.7%

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

   12. Simplified 66.7%

       \[\leadsto \sqrt{\ell} \cdot {\color{blue}{\left(\frac{-2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell} \cdot {\left(\frac{-2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(\frac{U*}{\frac{\frac{Om}{l\_m}}{n}} + l\_m \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.5e+108)
   (sqrt
    (*
     2.0
     (*
      (* n U)
      (+ t (* (/ l_m Om) (+ (/ U* (/ (/ Om l_m) n)) (* l_m -2.0)))))))
   (* (* l_m (sqrt 2.0)) (sqrt (/ (* (* n U) (+ -2.0 (/ (* n U*) Om))) 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 <= 5.5e+108) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((((n * U) * (-2.0 + ((n * U_42_) / 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) :: tmp
    if (l_m <= 5.5d+108) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m / om) * ((u_42 / ((om / l_m) / n)) + (l_m * (-2.0d0))))))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt((((n * u) * ((-2.0d0) + ((n * u_42) / 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 tmp;
	if (l_m <= 5.5e+108) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.5e+108:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.5e+108)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m / Om) * Float64(Float64(U_42_ / Float64(Float64(Om / l_m) / n)) + Float64(l_m * -2.0)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(Float64(n * U) * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))) / 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 <= 5.5e+108)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / 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, 5.5e+108], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ / N[(N[(Om / l$95$m), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.4999999999999998e108

   1. Initial program 55.4%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 58.3%

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

   7. Simplified 58.3%

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

      1. *-commutative N/A

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

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

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

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

      7. /-lowering-/.f64 59.8%

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

   9. Applied egg-rr 59.8%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

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

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

       6. associate-/r* N/A

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

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

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

       8. /-lowering-/.f64 59.8%

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

   11. Applied egg-rr 59.8%

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

   if 5.4999999999999998e108 < l

   1. Initial program 16.6%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 46.0%

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

   7. Simplified 46.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   9. Applied egg-rr 43.8%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       11. *-lowering-*.f64 45.5%

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

   12. Simplified 45.5%

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

   13. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\sqrt{\frac{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   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 50.3%

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

   15. Simplified 50.3%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;Om \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\_1\right)}\\ \mathbf{elif}\;Om \leq -1.75 \cdot 10^{-256}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(0 - n \cdot l\_m\right)}{Om}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (* l_m l_m) Om)))))
   (if (<= Om -4.5e-73)
     (sqrt (* (* 2.0 U) (* n t_1)))
     (if (<= Om -1.75e-256)
       (/ (* (sqrt (* 2.0 (* U (- U* U)))) (- 0.0 (* n l_m))) Om)
       (if (<= Om 4e-114)
         (sqrt (* n (/ (* 2.0 (* U (* U* (* n (* l_m l_m))))) (* Om Om))))
         (sqrt (* (* (* 2.0 n) U) 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 = t - (2.0 * ((l_m * l_m) / Om));
	double tmp;
	if (Om <= -4.5e-73) {
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	} else if (Om <= -1.75e-256) {
		tmp = (sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om;
	} else if (Om <= 4e-114) {
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * 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 - (2.0d0 * ((l_m * l_m) / om))
    if (om <= (-4.5d-73)) then
        tmp = sqrt(((2.0d0 * u) * (n * t_1)))
    else if (om <= (-1.75d-256)) then
        tmp = (sqrt((2.0d0 * (u * (u_42 - u)))) * (0.0d0 - (n * l_m))) / om
    else if (om <= 4d-114) then
        tmp = sqrt((n * ((2.0d0 * (u * (u_42 * (n * (l_m * l_m))))) / (om * om))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * 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 = t - (2.0 * ((l_m * l_m) / Om));
	double tmp;
	if (Om <= -4.5e-73) {
		tmp = Math.sqrt(((2.0 * U) * (n * t_1)));
	} else if (Om <= -1.75e-256) {
		tmp = (Math.sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om;
	} else if (Om <= 4e-114) {
		tmp = Math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t - (2.0 * ((l_m * l_m) / Om))
	tmp = 0
	if Om <= -4.5e-73:
		tmp = math.sqrt(((2.0 * U) * (n * t_1)))
	elif Om <= -1.75e-256:
		tmp = (math.sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om
	elif Om <= 4e-114:
		tmp = math.sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om)))
	tmp = 0.0
	if (Om <= -4.5e-73)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t_1)));
	elseif (Om <= -1.75e-256)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(U * Float64(U_42_ - U)))) * Float64(0.0 - Float64(n * l_m))) / Om);
	elseif (Om <= 4e-114)
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(U_42_ * Float64(n * Float64(l_m * l_m))))) / Float64(Om * Om))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * 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 - (2.0 * ((l_m * l_m) / Om));
	tmp = 0.0;
	if (Om <= -4.5e-73)
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	elseif (Om <= -1.75e-256)
		tmp = (sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om;
	elseif (Om <= 4e-114)
		tmp = sqrt((n * ((2.0 * (U * (U_42_ * (n * (l_m * l_m))))) / (Om * Om))));
	else
		tmp = sqrt((((2.0 * n) * U) * 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[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -4.5e-73], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.75e-256], N[(N[(N[Sqrt[N[(2.0 * N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], If[LessEqual[Om, 4e-114], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -4.5e-73

   1. Initial program 43.0%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 43.5%

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

   7. Simplified 43.5%

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

   if -4.5e-73 < Om < -1.75000000000000007e-256

   1. Initial program 61.1%

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

   3. Simplified 57.9%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

   6. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot n\right)\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 22.9%

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

   9. Simplified 22.9%

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

       1. associate-*l/ N/A

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

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

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sqrt-unprod N/A

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

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

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

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

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

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

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

       11. --lowering--.f64 23.0%

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

   11. Applied egg-rr 23.0%

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

   if -1.75000000000000007e-256 < Om < 4.0000000000000002e-114

   1. Initial program 43.3%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 60.6%

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

   7. Simplified 60.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   9. Applied egg-rr 58.7%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       11. *-lowering-*.f64 52.7%

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

   12. Simplified 52.7%

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

   13. Taylor expanded in U* around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 39.8%

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

   15. Simplified 39.8%

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

   if 4.0000000000000002e-114 < Om

   1. Initial program 56.4%

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

   3. Taylor expanded in Om around inf

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -1.75 \cdot 10^{-256}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(0 - n \cdot \ell\right)}{Om}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;Om \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\_1\right)}\\ \mathbf{elif}\;Om \leq -1.12 \cdot 10^{-298}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(0 - n \cdot l\_m\right)}{Om}\\ \mathbf{elif}\;Om \leq 5.3 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \frac{\left(\left(l\_m \cdot l\_m\right) \cdot U*\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (* l_m l_m) Om)))))
   (if (<= Om -4.5e-73)
     (sqrt (* (* 2.0 U) (* n t_1)))
     (if (<= Om -1.12e-298)
       (/ (* (sqrt (* 2.0 (* U (- U* U)))) (- 0.0 (* n l_m))) Om)
       (if (<= Om 5.3e-202)
         (sqrt (* 2.0 (* U (/ (* (* (* l_m l_m) U*) (* n n)) (* Om Om)))))
         (sqrt (* (* (* 2.0 n) U) 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 = t - (2.0 * ((l_m * l_m) / Om));
	double tmp;
	if (Om <= -4.5e-73) {
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	} else if (Om <= -1.12e-298) {
		tmp = (sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om;
	} else if (Om <= 5.3e-202) {
		tmp = sqrt((2.0 * (U * ((((l_m * l_m) * U_42_) * (n * n)) / (Om * Om)))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * 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 - (2.0d0 * ((l_m * l_m) / om))
    if (om <= (-4.5d-73)) then
        tmp = sqrt(((2.0d0 * u) * (n * t_1)))
    else if (om <= (-1.12d-298)) then
        tmp = (sqrt((2.0d0 * (u * (u_42 - u)))) * (0.0d0 - (n * l_m))) / om
    else if (om <= 5.3d-202) then
        tmp = sqrt((2.0d0 * (u * ((((l_m * l_m) * u_42) * (n * n)) / (om * om)))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * 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 = t - (2.0 * ((l_m * l_m) / Om));
	double tmp;
	if (Om <= -4.5e-73) {
		tmp = Math.sqrt(((2.0 * U) * (n * t_1)));
	} else if (Om <= -1.12e-298) {
		tmp = (Math.sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om;
	} else if (Om <= 5.3e-202) {
		tmp = Math.sqrt((2.0 * (U * ((((l_m * l_m) * U_42_) * (n * n)) / (Om * Om)))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t - (2.0 * ((l_m * l_m) / Om))
	tmp = 0
	if Om <= -4.5e-73:
		tmp = math.sqrt(((2.0 * U) * (n * t_1)))
	elif Om <= -1.12e-298:
		tmp = (math.sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om
	elif Om <= 5.3e-202:
		tmp = math.sqrt((2.0 * (U * ((((l_m * l_m) * U_42_) * (n * n)) / (Om * Om)))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om)))
	tmp = 0.0
	if (Om <= -4.5e-73)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t_1)));
	elseif (Om <= -1.12e-298)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(U * Float64(U_42_ - U)))) * Float64(0.0 - Float64(n * l_m))) / Om);
	elseif (Om <= 5.3e-202)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(Float64(Float64(l_m * l_m) * U_42_) * Float64(n * n)) / Float64(Om * Om)))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * 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 - (2.0 * ((l_m * l_m) / Om));
	tmp = 0.0;
	if (Om <= -4.5e-73)
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	elseif (Om <= -1.12e-298)
		tmp = (sqrt((2.0 * (U * (U_42_ - U)))) * (0.0 - (n * l_m))) / Om;
	elseif (Om <= 5.3e-202)
		tmp = sqrt((2.0 * (U * ((((l_m * l_m) * U_42_) * (n * n)) / (Om * Om)))));
	else
		tmp = sqrt((((2.0 * n) * U) * 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[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -4.5e-73], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.12e-298], N[(N[(N[Sqrt[N[(2.0 * N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], If[LessEqual[Om, 5.3e-202], N[Sqrt[N[(2.0 * N[(U * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -4.5e-73

   1. Initial program 43.0%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 43.5%

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

   7. Simplified 43.5%

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

   if -4.5e-73 < Om < -1.1200000000000001e-298

   1. Initial program 56.8%

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

   3. Simplified 56.6%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

   6. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(U \cdot n\right)\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. --lowering--.f64 22.4%

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

   9. Simplified 22.4%

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

       1. associate-*l/ N/A

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

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

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sqrt-unprod N/A

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

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

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

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

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

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

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

       11. --lowering--.f64 22.5%

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

   11. Applied egg-rr 22.5%

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

   if -1.1200000000000001e-298 < Om < 5.30000000000000042e-202

   1. Initial program 22.8%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in U* around inf

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 50.9%

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

   7. Simplified 50.9%

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

   if 5.30000000000000042e-202 < Om

   1. Initial program 56.9%

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

   3. Taylor expanded in Om around inf

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 47.4%

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

   5. Simplified 47.4%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -1.12 \cdot 10^{-298}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)} \cdot \left(0 - n \cdot \ell\right)}{Om}\\ \mathbf{elif}\;Om \leq 5.3 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -5.2 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right) + -4 \cdot \frac{U \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 5.8 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(n \cdot l\_m\right) \cdot \left(l\_m \cdot -2 + \frac{U* \cdot \left(n \cdot l\_m\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -5.2e-66)
   (sqrt (+ (* (* n t) (* 2.0 U)) (* -4.0 (/ (* U (* n (* l_m l_m))) Om))))
   (if (<= Om 5.8e-111)
     (sqrt
      (/
       (* 2.0 (* U (* (* n l_m) (+ (* l_m -2.0) (/ (* U* (* n l_m)) Om)))))
       Om))
     (sqrt (* (* (* 2.0 n) U) (- 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 tmp;
	if (Om <= -5.2e-66) {
		tmp = sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))));
	} else if (Om <= 5.8e-111) {
		tmp = sqrt(((2.0 * (U * ((n * l_m) * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (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) :: tmp
    if (om <= (-5.2d-66)) then
        tmp = sqrt((((n * t) * (2.0d0 * u)) + ((-4.0d0) * ((u * (n * (l_m * l_m))) / om))))
    else if (om <= 5.8d-111) then
        tmp = sqrt(((2.0d0 * (u * ((n * l_m) * ((l_m * (-2.0d0)) + ((u_42 * (n * l_m)) / om))))) / om))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (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 tmp;
	if (Om <= -5.2e-66) {
		tmp = Math.sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))));
	} else if (Om <= 5.8e-111) {
		tmp = Math.sqrt(((2.0 * (U * ((n * l_m) * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (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_):
	tmp = 0
	if Om <= -5.2e-66:
		tmp = math.sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))))
	elif Om <= 5.8e-111:
		tmp = math.sqrt(((2.0 * (U * ((n * l_m) * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (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_)
	tmp = 0.0
	if (Om <= -5.2e-66)
		tmp = sqrt(Float64(Float64(Float64(n * t) * Float64(2.0 * U)) + Float64(-4.0 * Float64(Float64(U * Float64(n * Float64(l_m * l_m))) / Om))));
	elseif (Om <= 5.8e-111)
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(n * l_m) * Float64(Float64(l_m * -2.0) + Float64(Float64(U_42_ * Float64(n * l_m)) / Om))))) / Om));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(2.0 * Float64(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_)
	tmp = 0.0;
	if (Om <= -5.2e-66)
		tmp = sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))));
	elseif (Om <= 5.8e-111)
		tmp = sqrt(((2.0 * (U * ((n * l_m) * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om));
	else
		tmp = sqrt((((2.0 * n) * U) * (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$42$_] := If[LessEqual[Om, -5.2e-66], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 5.8e-111], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(n * l$95$m), $MachinePrecision] * N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(U$42$ * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -5.1999999999999998e-66

   1. Initial program 43.0%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. 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. *-lowering-*.f64 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 44.6%

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

   7. Simplified 44.6%

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

   if -5.1999999999999998e-66 < Om < 5.80000000000000003e-111

   1. Initial program 50.4%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 59.5%

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

   7. Simplified 59.5%

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

      1. *-commutative N/A

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

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

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

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

      7. /-lowering-/.f64 60.9%

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

   9. Applied egg-rr 60.9%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 60.0%

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

   12. Simplified 60.0%

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

   if 5.80000000000000003e-111 < Om

   1. Initial program 56.4%

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

   3. Taylor expanded in Om around inf

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -5.2 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right) + -4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 5.8 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right) + -4 \cdot \frac{U \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 3.3 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U \cdot \left(l\_m \cdot \left(n \cdot \left(l\_m \cdot -2 + \frac{U* \cdot \left(n \cdot l\_m\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -4.7e-71)
   (sqrt (+ (* (* n t) (* 2.0 U)) (* -4.0 (/ (* U (* n (* l_m l_m))) Om))))
   (if (<= Om 3.3e-110)
     (sqrt
      (*
       2.0
       (/ (* U (* l_m (* n (+ (* l_m -2.0) (/ (* U* (* n l_m)) Om))))) Om)))
     (sqrt (* (* (* 2.0 n) U) (- 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 tmp;
	if (Om <= -4.7e-71) {
		tmp = sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))));
	} else if (Om <= 3.3e-110) {
		tmp = sqrt((2.0 * ((U * (l_m * (n * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (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) :: tmp
    if (om <= (-4.7d-71)) then
        tmp = sqrt((((n * t) * (2.0d0 * u)) + ((-4.0d0) * ((u * (n * (l_m * l_m))) / om))))
    else if (om <= 3.3d-110) then
        tmp = sqrt((2.0d0 * ((u * (l_m * (n * ((l_m * (-2.0d0)) + ((u_42 * (n * l_m)) / om))))) / om)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (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 tmp;
	if (Om <= -4.7e-71) {
		tmp = Math.sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))));
	} else if (Om <= 3.3e-110) {
		tmp = Math.sqrt((2.0 * ((U * (l_m * (n * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (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_):
	tmp = 0
	if Om <= -4.7e-71:
		tmp = math.sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))))
	elif Om <= 3.3e-110:
		tmp = math.sqrt((2.0 * ((U * (l_m * (n * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (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_)
	tmp = 0.0
	if (Om <= -4.7e-71)
		tmp = sqrt(Float64(Float64(Float64(n * t) * Float64(2.0 * U)) + Float64(-4.0 * Float64(Float64(U * Float64(n * Float64(l_m * l_m))) / Om))));
	elseif (Om <= 3.3e-110)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * Float64(l_m * Float64(n * Float64(Float64(l_m * -2.0) + Float64(Float64(U_42_ * Float64(n * l_m)) / Om))))) / Om)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(2.0 * Float64(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_)
	tmp = 0.0;
	if (Om <= -4.7e-71)
		tmp = sqrt((((n * t) * (2.0 * U)) + (-4.0 * ((U * (n * (l_m * l_m))) / Om))));
	elseif (Om <= 3.3e-110)
		tmp = sqrt((2.0 * ((U * (l_m * (n * ((l_m * -2.0) + ((U_42_ * (n * l_m)) / Om))))) / Om)));
	else
		tmp = sqrt((((2.0 * n) * U) * (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$42$_] := If[LessEqual[Om, -4.7e-71], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 3.3e-110], N[Sqrt[N[(2.0 * N[(N[(U * N[(l$95$m * N[(n * N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(U$42$ * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -4.69999999999999996e-71

   1. Initial program 43.0%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. 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. *-lowering-*.f64 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 44.6%

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

   7. Simplified 44.6%

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

   if -4.69999999999999996e-71 < Om < 3.2999999999999999e-110

   1. Initial program 50.4%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 59.0%

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

   10. Simplified 59.0%

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

   if 3.2999999999999999e-110 < Om

   1. Initial program 56.4%

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

   3. Taylor expanded in Om around inf

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.7 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right) + -4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 3.3 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(\ell \cdot -2 + \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.96 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(t + \frac{\frac{U*}{\frac{\frac{Om}{l\_m}}{n}} + l\_m \cdot -2}{\frac{Om}{l\_m}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot l\_m\right)\right) \cdot \frac{l\_m \cdot -2 + \frac{n \cdot U*}{\frac{Om}{l\_m}}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.96e+177)
   (sqrt
    (*
     (+ t (/ (+ (/ U* (/ (/ Om l_m) n)) (* l_m -2.0)) (/ Om l_m)))
     (* 2.0 (* n U))))
   (sqrt
    (*
     n
     (* (* 2.0 (* U l_m)) (/ (+ (* l_m -2.0) (/ (* n U*) (/ Om 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 tmp;
	if (l_m <= 1.96e+177) {
		tmp = sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / 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) :: tmp
    if (l_m <= 1.96d+177) then
        tmp = sqrt(((t + (((u_42 / ((om / l_m) / n)) + (l_m * (-2.0d0))) / (om / l_m))) * (2.0d0 * (n * u))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * l_m)) * (((l_m * (-2.0d0)) + ((n * u_42) / (om / 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 tmp;
	if (l_m <= 1.96e+177) {
		tmp = Math.sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.96e+177:
		tmp = math.sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.96e+177)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(U_42_ / Float64(Float64(Om / l_m) / n)) + Float64(l_m * -2.0)) / Float64(Om / l_m))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * l_m)) * Float64(Float64(Float64(l_m * -2.0) + Float64(Float64(n * U_42_) / Float64(Om / l_m))) / 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 <= 1.96e+177)
		tmp = sqrt(((t + (((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)) / (Om / l_m))) * (2.0 * (n * U))));
	else
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.96e+177], N[Sqrt[N[(N[(t + N[(N[(N[(U$42$ / N[(N[(Om / l$95$m), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(n * U$42$), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.96e177

   1. Initial program 53.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 56.7%

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

   7. Simplified 56.7%

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

      1. *-commutative N/A

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

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

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

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

      7. /-lowering-/.f64 58.0%

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

   9. Applied egg-rr 58.0%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

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

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

   11. Applied egg-rr 58.1%

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

   if 1.96e177 < l

   1. Initial program 12.8%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 56.6%

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

   7. Simplified 56.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   9. Applied egg-rr 57.4%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       11. *-lowering-*.f64 57.4%

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

   12. Simplified 57.4%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

       3. associate-/l* N/A

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

       4. clear-num N/A

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

       5. associate-/r* N/A

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

       6. div-inv N/A

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

       7. *-commutative N/A

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

       8. +-commutative N/A

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

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \mathsf{/.f64}\left(\left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right), Om\right)\right), n\right)\right) \]

   14. Applied egg-rr 67.7%

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U* \cdot n}{\frac{Om}{\ell}}}{Om}\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.96 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{n \cdot U*}{\frac{Om}{\ell}}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.75 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(\frac{U*}{\frac{\frac{Om}{l\_m}}{n}} + l\_m \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot l\_m\right)\right) \cdot \frac{l\_m \cdot -2 + \frac{n \cdot U*}{\frac{Om}{l\_m}}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.75e+177)
   (sqrt
    (*
     2.0
     (*
      (* n U)
      (+ t (* (/ l_m Om) (+ (/ U* (/ (/ Om l_m) n)) (* l_m -2.0)))))))
   (sqrt
    (*
     n
     (* (* 2.0 (* U l_m)) (/ (+ (* l_m -2.0) (/ (* n U*) (/ Om 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 tmp;
	if (l_m <= 1.75e+177) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / 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) :: tmp
    if (l_m <= 1.75d+177) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m / om) * ((u_42 / ((om / l_m) / n)) + (l_m * (-2.0d0))))))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * l_m)) * (((l_m * (-2.0d0)) + ((n * u_42) / (om / 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 tmp;
	if (l_m <= 1.75e+177) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.75e+177:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.75e+177)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m / Om) * Float64(Float64(U_42_ / Float64(Float64(Om / l_m) / n)) + Float64(l_m * -2.0)))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * l_m)) * Float64(Float64(Float64(l_m * -2.0) + Float64(Float64(n * U_42_) / Float64(Om / l_m))) / 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 <= 1.75e+177)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((U_42_ / ((Om / l_m) / n)) + (l_m * -2.0)))))));
	else
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.75e+177], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ / N[(N[(Om / l$95$m), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(n * U$42$), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.74999999999999996e177

   1. Initial program 53.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 56.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(\frac{\ell \cdot n}{Om} \cdot U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell \cdot n}{Om}\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{n}{Om}\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{Om} \cdot \ell\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      5. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{\frac{Om}{\ell}}\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      7. /-lowering-/.f64 58.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   9. Applied egg-rr 58.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{\frac{n}{\frac{Om}{\ell}} \cdot U*} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{n}{\frac{Om}{\ell}}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{1}{\frac{\frac{Om}{\ell}}{n}}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{1}{\frac{Om}{\ell \cdot n}}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(\frac{U*}{\frac{Om}{\ell \cdot n}}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(U*, \left(\frac{Om}{\ell \cdot n}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(U*, \left(\frac{\frac{Om}{\ell}}{n}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(U*, \mathsf{/.f64}\left(\left(\frac{Om}{\ell}\right), n\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

       8. /-lowering-/.f64 58.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{/.f64}\left(Om, \ell\right), n\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   11. Applied egg-rr 58.1%

       \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]

   if 1.74999999999999996e177 < l

   1. Initial program 12.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 56.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   9. Applied egg-rr 57.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U*}{\frac{Om}{\ell \cdot n}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       11. *-lowering-*.f64 57.4%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 57.4%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}} \cdot n} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right), n\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}}{Om}\right), n\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right), n\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{1}{\frac{Om}{\ell \cdot n}}}{Om}\right), n\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{1}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       6. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + \frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right), n\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \mathsf{/.f64}\left(\left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right), Om\right)\right), n\right)\right) \]

   14. Applied egg-rr 67.7%

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U* \cdot n}{\frac{Om}{\ell}}}{Om}\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{n \cdot U*}{\frac{Om}{\ell}}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.9 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(l\_m \cdot -2 + U* \cdot \frac{n}{\frac{Om}{l\_m}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot l\_m\right)\right) \cdot \frac{l\_m \cdot -2 + \frac{n \cdot U*}{\frac{Om}{l\_m}}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.9e+177)
   (sqrt
    (*
     2.0
     (*
      (* n U)
      (+ t (* (/ l_m Om) (+ (* l_m -2.0) (* U* (/ n (/ Om l_m)))))))))
   (sqrt
    (*
     n
     (* (* 2.0 (* U l_m)) (/ (+ (* l_m -2.0) (/ (* n U*) (/ Om 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 tmp;
	if (l_m <= 2.9e+177) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * (n / (Om / l_m)))))))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / 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) :: tmp
    if (l_m <= 2.9d+177) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m / om) * ((l_m * (-2.0d0)) + (u_42 * (n / (om / l_m)))))))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * l_m)) * (((l_m * (-2.0d0)) + ((n * u_42) / (om / 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 tmp;
	if (l_m <= 2.9e+177) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * (n / (Om / l_m)))))))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.9e+177:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * (n / (Om / l_m)))))))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.9e+177)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l_m / Om) * Float64(Float64(l_m * -2.0) + Float64(U_42_ * Float64(n / Float64(Om / l_m)))))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * l_m)) * Float64(Float64(Float64(l_m * -2.0) + Float64(Float64(n * U_42_) / Float64(Om / l_m))) / 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 <= 2.9e+177)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * (n / (Om / l_m)))))))));
	else
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.9e+177], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(U$42$ * N[(n / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(n * U$42$), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.90000000000000013e177

   1. Initial program 53.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 56.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(\frac{\ell \cdot n}{Om} \cdot U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell \cdot n}{Om}\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{n}{Om}\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{Om} \cdot \ell\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      5. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{\frac{Om}{\ell}}\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      7. /-lowering-/.f64 58.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right), U*\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   9. Applied egg-rr 58.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{\frac{n}{\frac{Om}{\ell}} \cdot U*} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]

   if 2.90000000000000013e177 < l

   1. Initial program 12.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 56.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   9. Applied egg-rr 57.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U*}{\frac{Om}{\ell \cdot n}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       11. *-lowering-*.f64 57.4%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 57.4%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}} \cdot n} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right), n\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}}{Om}\right), n\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right), n\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{1}{\frac{Om}{\ell \cdot n}}}{Om}\right), n\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{1}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       6. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + \frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right), n\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \mathsf{/.f64}\left(\left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right), Om\right)\right), n\right)\right) \]

   14. Applied egg-rr 67.7%

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U* \cdot n}{\frac{Om}{\ell}}}{Om}\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{n \cdot U*}{\frac{Om}{\ell}}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.65 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{l\_m}{Om} \cdot \left(l\_m \cdot -2 + U* \cdot \frac{n \cdot l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot l\_m\right)\right) \cdot \frac{l\_m \cdot -2 + \frac{n \cdot U*}{\frac{Om}{l\_m}}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.65e+177)
   (sqrt
    (*
     2.0
     (*
      (* n U)
      (+ t (* (/ l_m Om) (+ (* l_m -2.0) (* U* (/ (* n l_m) Om))))))))
   (sqrt
    (*
     n
     (* (* 2.0 (* U l_m)) (/ (+ (* l_m -2.0) (/ (* n U*) (/ Om 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 tmp;
	if (l_m <= 1.65e+177) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om))))))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / 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) :: tmp
    if (l_m <= 1.65d+177) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l_m / om) * ((l_m * (-2.0d0)) + (u_42 * ((n * l_m) / om))))))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * l_m)) * (((l_m * (-2.0d0)) + ((n * u_42) / (om / 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 tmp;
	if (l_m <= 1.65e+177) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om))))))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.65e+177:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om))))))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.65e+177)
		tmp = sqrt(Float64(2.0 * Float64(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))))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * l_m)) * Float64(Float64(Float64(l_m * -2.0) + Float64(Float64(n * U_42_) / Float64(Om / l_m))) / 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 <= 1.65e+177)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l_m / Om) * ((l_m * -2.0) + (U_42_ * ((n * l_m) / Om))))))));
	else
		tmp = sqrt((n * ((2.0 * (U * l_m)) * (((l_m * -2.0) + ((n * U_42_) / (Om / l_m))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.65e+177], N[Sqrt[N[(2.0 * N[(N[(n * U), $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]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * -2.0), $MachinePrecision] + N[(N[(n * U$42$), $MachinePrecision] / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6500000000000001e177

   1. Initial program 53.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 56.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]

   if 1.6500000000000001e177 < l

   1. Initial program 12.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 56.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   9. Applied egg-rr 57.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U*}{\frac{Om}{\ell \cdot n}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       11. *-lowering-*.f64 57.4%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 57.4%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}} \cdot n} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right), n\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}}{Om}\right), n\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right), n\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{1}{\frac{Om}{\ell \cdot n}}}{Om}\right), n\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{1}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       6. div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + \frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{Om}\right), n\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right), n\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \ell\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \left(\frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{Om}\right)\right), n\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, U\right)\right), \mathsf{/.f64}\left(\left(\frac{U*}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2\right), Om\right)\right), n\right)\right) \]

   14. Applied egg-rr 67.7%

       \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \frac{\ell \cdot -2 + \frac{U* \cdot n}{\frac{Om}{\ell}}}{Om}\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \frac{n \cdot U*}{\frac{Om}{\ell}}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{-30}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.85e-30)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* -2.0 (/ (* U (* (* l_m l_m) (* n (- 2.0 (/ (* n U*) Om))))) 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 <= 1.85e-30) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((-2.0 * ((U * ((l_m * l_m) * (n * (2.0 - ((n * U_42_) / 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) :: tmp
    if (l_m <= 1.85d-30) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((-2.0d0) * ((u * ((l_m * l_m) * (n * (2.0d0 - ((n * u_42) / 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 tmp;
	if (l_m <= 1.85e-30) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * ((U * ((l_m * l_m) * (n * (2.0 - ((n * U_42_) / Om))))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.85e-30:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * ((U * ((l_m * l_m) * (n * (2.0 - ((n * U_42_) / Om))))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.85e-30)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(2.0 - Float64(Float64(n * U_42_) / Om))))) / 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 <= 1.85e-30)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * ((U * ((l_m * l_m) * (n * (2.0 - ((n * U_42_) / Om))))) / 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, 1.85e-30], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8500000000000002e-30

   1. Initial program 55.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 36.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 36.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} \cdot \color{blue}{2}\right)} \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\left(\frac{1}{4} \cdot 2\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      10. metadata-eval 39.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 39.7%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

   if 1.8500000000000002e-30 < l

   1. Initial program 29.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 48.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 48.3%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]

   8. Taylor expanded in l around -inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)}{Om}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(n \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, \left(2 + \left(\mathsf{neg}\left(\frac{U* \cdot n}{Om}\right)\right)\right)\right)\right)\right), Om\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(2, \left(\frac{U* \cdot n}{Om}\right)\right)\right)\right)\right), Om\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(U* \cdot n\right), Om\right)\right)\right)\right)\right), Om\right)\right)\right) \]

      12. *-lowering-*.f64 39.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, n\right), Om\right)\right)\right)\right)\right), Om\right)\right)\right) \]

   10. Simplified 39.0%

       \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-30}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{\left(n \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(2 \cdot U*\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.3e+175)
   (sqrt (* (* (* 2.0 n) U) (- t (* 2.0 (/ (* l_m l_m) Om)))))
   (sqrt (* n (* U (/ (* (* n (* l_m l_m)) (* 2.0 U*)) (* Om 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 <= 1.3e+175) {
		tmp = sqrt((((2.0 * n) * U) * (t - (2.0 * ((l_m * l_m) / Om)))));
	} else {
		tmp = sqrt((n * (U * (((n * (l_m * l_m)) * (2.0 * U_42_)) / (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) :: tmp
    if (l_m <= 1.3d+175) then
        tmp = sqrt((((2.0d0 * n) * u) * (t - (2.0d0 * ((l_m * l_m) / om)))))
    else
        tmp = sqrt((n * (u * (((n * (l_m * l_m)) * (2.0d0 * u_42)) / (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 tmp;
	if (l_m <= 1.3e+175) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - (2.0 * ((l_m * l_m) / Om)))));
	} else {
		tmp = Math.sqrt((n * (U * (((n * (l_m * l_m)) * (2.0 * U_42_)) / (Om * Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.3e+175:
		tmp = math.sqrt((((2.0 * n) * U) * (t - (2.0 * ((l_m * l_m) / Om)))))
	else:
		tmp = math.sqrt((n * (U * (((n * (l_m * l_m)) * (2.0 * U_42_)) / (Om * Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.3e+175)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om)))));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(l_m * l_m)) * Float64(2.0 * U_42_)) / Float64(Om * 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 <= 1.3e+175)
		tmp = sqrt((((2.0 * n) * U) * (t - (2.0 * ((l_m * l_m) / Om)))));
	else
		tmp = sqrt((n * (U * (((n * (l_m * l_m)) * (2.0 * U_42_)) / (Om * 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, 1.3e+175], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e175

   1. Initial program 53.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right) \]

   5. Simplified 41.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   if 1.3e175 < l

   1. Initial program 12.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 54.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   9. Applied egg-rr 54.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U*}{\frac{Om}{\ell \cdot n}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   10. Taylor expanded in U* around inf

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}, U\right), n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right), U\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot U*\right) \cdot \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U*\right), \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U*\right), \left({\ell}^{2} \cdot n\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U*\right), \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U*\right), \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U*\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left({Om}^{2}\right)\right), U\right), n\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U*\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(Om \cdot Om\right)\right), U\right), n\right)\right) \]

       10. *-lowering-*.f64 26.7%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U*\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), U\right), n\right)\right) \]

   12. Simplified 26.7%

       \[\leadsto \sqrt{\left(\color{blue}{\frac{\left(2 \cdot U*\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om}} \cdot U\right) \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 \cdot U*\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 9 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 9e+174)
     (sqrt (* t_1 (- t (* 2.0 (/ (* l_m l_m) Om)))))
     (sqrt (* t_1 (/ (* 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 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 9e+174) {
		tmp = sqrt((t_1 * (t - (2.0 * ((l_m * l_m) / Om)))));
	} else {
		tmp = sqrt((t_1 * ((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 = (2.0d0 * n) * u
    if (l_m <= 9d+174) then
        tmp = sqrt((t_1 * (t - (2.0d0 * ((l_m * l_m) / om)))))
    else
        tmp = sqrt((t_1 * ((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 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 9e+174) {
		tmp = Math.sqrt((t_1 * (t - (2.0 * ((l_m * l_m) / Om)))));
	} else {
		tmp = Math.sqrt((t_1 * ((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 = (2.0 * n) * U
	tmp = 0
	if l_m <= 9e+174:
		tmp = math.sqrt((t_1 * (t - (2.0 * ((l_m * l_m) / Om)))))
	else:
		tmp = math.sqrt((t_1 * ((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(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 9e+174)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om)))));
	else
		tmp = sqrt(Float64(t_1 * 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 = (2.0 * n) * U;
	tmp = 0.0;
	if (l_m <= 9e+174)
		tmp = sqrt((t_1 * (t - (2.0 * ((l_m * l_m) / Om)))));
	else
		tmp = sqrt((t_1 * ((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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 9e+174], N[Sqrt[N[(t$95$1 * N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * 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 2 regimes

2. if l < 9.00000000000000084e174

   1. Initial program 53.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 41.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right) \]

   5. Simplified 41.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   if 9.00000000000000084e174 < l

   1. Initial program 12.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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 26.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right) \]

   5. Simplified 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;Om \leq 1.75 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (* l_m l_m) Om)))))
   (if (<= Om 1.75e-114)
     (sqrt (* (* 2.0 U) (* n t_1)))
     (sqrt (* (* (* 2.0 n) U) 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 = t - (2.0 * ((l_m * l_m) / Om));
	double tmp;
	if (Om <= 1.75e-114) {
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * 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 - (2.0d0 * ((l_m * l_m) / om))
    if (om <= 1.75d-114) then
        tmp = sqrt(((2.0d0 * u) * (n * t_1)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * 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 = t - (2.0 * ((l_m * l_m) / Om));
	double tmp;
	if (Om <= 1.75e-114) {
		tmp = Math.sqrt(((2.0 * U) * (n * t_1)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t - (2.0 * ((l_m * l_m) / Om))
	tmp = 0
	if Om <= 1.75e-114:
		tmp = math.sqrt(((2.0 * U) * (n * t_1)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om)))
	tmp = 0.0
	if (Om <= 1.75e-114)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t_1)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * 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 - (2.0 * ((l_m * l_m) / Om));
	tmp = 0.0;
	if (Om <= 1.75e-114)
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	else
		tmp = sqrt((((2.0 * n) * U) * 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[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, 1.75e-114], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.75e-114

   1. Initial program 47.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 34.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 34.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   if 1.75e-114 < Om

   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. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 51.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right) \]

   5. Simplified 51.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.25e-12)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* (* 2.0 U) (* n (- 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 tmp;
	if (l_m <= 1.25e-12) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt(((2.0 * U) * (n * (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) :: tmp
    if (l_m <= 1.25d-12) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * u) * (n * (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 tmp;
	if (l_m <= 1.25e-12) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (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_):
	tmp = 0
	if l_m <= 1.25e-12:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (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_)
	tmp = 0.0
	if (l_m <= 1.25e-12)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(2.0 * Float64(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_)
	tmp = 0.0;
	if (l_m <= 1.25e-12)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * U) * (n * (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$42$_] := If[LessEqual[l$95$m, 1.25e-12], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.24999999999999992e-12

   1. Initial program 55.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 35.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 35.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} \cdot \color{blue}{2}\right)} \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\left(\frac{1}{4} \cdot 2\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      10. metadata-eval 39.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 39.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

   if 1.24999999999999992e-12 < l

   1. Initial program 26.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 46.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 21.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 21.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{+56}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(U \cdot -4\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.6e+56)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (/ (* (* n (* l_m l_m)) (* U -4.0)) 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 <= 5.6e+56) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((((n * (l_m * l_m)) * (U * -4.0)) / 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 <= 5.6d+56) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt((((n * (l_m * l_m)) * (u * (-4.0d0))) / 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 <= 5.6e+56) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((((n * (l_m * l_m)) * (U * -4.0)) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.6e+56:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((((n * (l_m * l_m)) * (U * -4.0)) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.6e+56)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(Float64(n * Float64(l_m * l_m)) * Float64(U * -4.0)) / 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 <= 5.6e+56)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((((n * (l_m * l_m)) * (U * -4.0)) / 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, 5.6e+56], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.60000000000000017e56

   1. Initial program 56.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 35.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 35.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} \cdot \color{blue}{2}\right)} \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\left(\frac{1}{4} \cdot 2\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      10. metadata-eval 39.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 39.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

   if 5.60000000000000017e56 < l

   1. Initial program 19.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 44.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 44.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   9. Applied egg-rr 42.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U*}{\frac{Om}{\ell \cdot n}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       11. *-lowering-*.f64 44.3%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 44.3%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}} \cdot n} \]

   13. Taylor expanded in U* around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-4 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-4 \cdot U\right), \left({\ell}^{2} \cdot n\right)\right), Om\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, U\right), \left({\ell}^{2} \cdot n\right)\right), Om\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, U\right), \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), Om\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, U\right), \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), Om\right)\right) \]

       8. *-lowering-*.f64 19.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), Om\right)\right) \]

   15. Simplified 19.1%

       \[\leadsto \sqrt{\color{blue}{\frac{\left(-4 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+56}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot -4\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.1 \cdot 10^{+54}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.1e+54)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* n (/ (* -4.0 (* U (* 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 tmp;
	if (l_m <= 2.1e+54) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((n * ((-4.0 * (U * (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) :: tmp
    if (l_m <= 2.1d+54) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt((n * (((-4.0d0) * (u * (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 tmp;
	if (l_m <= 2.1e+54) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((n * ((-4.0 * (U * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.1e+54:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((n * ((-4.0 * (U * (l_m * l_m))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.1e+54)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(n * Float64(Float64(-4.0 * Float64(U * 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_)
	tmp = 0.0;
	if (l_m <= 2.1e+54)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((n * ((-4.0 * (U * (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$42$_] := If[LessEqual[l$95$m, 2.1e+54], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(n * N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.09999999999999986e54

   1. Initial program 56.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 35.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 35.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} \cdot \color{blue}{2}\right)} \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\left(\frac{1}{4} \cdot 2\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

      10. metadata-eval 39.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 39.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

   if 2.09999999999999986e54 < l

   1. Initial program 19.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 44.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 44.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   9. Applied egg-rr 42.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U*}{\frac{Om}{\ell \cdot n}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

       11. *-lowering-*.f64 44.3%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 44.3%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}} \cdot n} \]

   13. Taylor expanded in U* around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}, n\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot {\ell}^{2}\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot {\ell}^{2}\right)\right), Om\right), n\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right)\right), Om\right), n\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 14.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right), n\right)\right) \]

   15. Simplified 14.5%

       \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+54}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-48}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 5e-48)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* (* (* 2.0 n) U) t))))

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 <= 5e-48) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((((2.0 * n) * U) * t));
	}
	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 <= 5d-48) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((((2.0d0 * n) * u) * t))
    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 <= 5e-48) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= 5e-48:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 5e-48)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	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 <= 5e-48)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((((2.0 * n) * U) * t));
	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, 5e-48], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 4.9999999999999999e-48

   1. Initial program 48.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 28.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 28.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot U\right) \cdot t\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), t\right), n\right)\right) \]

      5. *-lowering-*.f64 25.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), t\right), n\right)\right) \]

   9. Applied egg-rr 25.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto {\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right)}^{\color{blue}{\frac{1}{2}}} \]

       2. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right), \color{blue}{\frac{1}{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right), \frac{1}{2}\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right), \frac{1}{2}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(t \cdot n\right)\right)\right), \frac{1}{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t \cdot n\right)\right)\right), \frac{1}{2}\right) \]

       7. *-lowering-*.f64 31.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, n\right)\right)\right), \frac{1}{2}\right) \]

   11. Applied egg-rr 31.3%

       \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)}^{0.5}} \]

   if 4.9999999999999999e-48 < Om

   1. Initial program 55.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-48}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 2.5 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 2.5e-289)
   (sqrt (* n (* t (* 2.0 U))))
   (sqrt (* (* n t) (* 2.0 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 (U <= 2.5e-289) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = sqrt(((n * t) * (2.0 * 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 (u <= 2.5d-289) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    else
        tmp = sqrt(((n * t) * (2.0d0 * 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 (U <= 2.5e-289) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 2.5e-289:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	else:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 2.5e-289)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * 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 (U <= 2.5e-289)
		tmp = sqrt((n * (t * (2.0 * U))));
	else
		tmp = sqrt(((n * t) * (2.0 * 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[U, 2.5e-289], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 2.50000000000000014e-289

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 28.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 28.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot U\right) \cdot t\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), t\right), n\right)\right) \]

      5. *-lowering-*.f64 32.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), t\right), n\right)\right) \]

   9. Applied egg-rr 32.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}} \]

   if 2.50000000000000014e-289 < U

   1. Initial program 51.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 34.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 34.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.5 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 37.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ {\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* 2.0 (* t (* 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((2.0 * (t * (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 = (2.0d0 * (t * (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((2.0 * (t * (n * U))), 0.5);
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.pow((2.0 * (t * (n * U))), 0.5)

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = (2.0 * (t * (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[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 56.1%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
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. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 31.5%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

7. Simplified 31.5%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
8. Step-by-step derivation

   1. pow1/2 N/A

      \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

   2. metadata-eval N/A

      \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1}{4} \cdot \color{blue}{2}\right)} \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\left(\frac{1}{4} \cdot 2\right)}\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(U \cdot n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \left(\frac{1}{4} \cdot 2\right)\right) \]

   10. metadata-eval 34.3%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(U, n\right)\right)\right), \frac{1}{2}\right) \]

9. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)}^{0.5}} \]

10. Final simplification 34.3%

    \[\leadsto {\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5} \]
11. Add Preprocessing

Alternative 23: 35.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* 2.0 n) U) t)))

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 * n) * U) * t));
}

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 * n) * u) * t))
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 * n) * U) * t));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * t))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * t))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
4. Step-by-step derivation

5. Simplified 32.3%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
6. Add Preprocessing

Alternative 24: 35.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* n t) (* 2.0 U))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((n * t) * (2.0 * 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(((n * t) * (2.0d0 * 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(((n * t) * (2.0 * U)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((n * t) * (2.0 * U)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(n * t) * Float64(2.0 * U)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((n * t) * (2.0 * U)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 56.1%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
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. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 31.5%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

7. Simplified 31.5%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

8. Final simplification 31.5%

   \[\leadsto \sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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*))))))