Toniolo and Linder, Equation (13)

Percentage Accurate: 49.4% → 62.1%

Time: 22.0s

Alternatives: 21

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: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 3.05 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -3.6e-279)
   (sqrt
    (*
     (+ (/ U (/ (/ Om l) (+ (* l -2.0) (* (/ n (/ Om l)) (- U* U))))) (* U t))
     (* n 2.0)))
   (if (<= n 3.05e-114)
     (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
     (*
      (sqrt
       (*
        (* U 2.0)
        (+ t (/ (+ (* l -2.0) (/ (- U* U) (/ (/ Om l) n))) (/ Om l)))))
      (sqrt n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -3.6e-279) {
		tmp = sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)));
	} else if (n <= 3.05e-114) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = sqrt(((U * 2.0) * (t + (((l * -2.0) + ((U_42_ - U) / ((Om / l) / n))) / (Om / l))))) * sqrt(n);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-3.6d-279)) then
        tmp = sqrt((((u / ((om / l) / ((l * (-2.0d0)) + ((n / (om / l)) * (u_42 - u))))) + (u * t)) * (n * 2.0d0)))
    else if (n <= 3.05d-114) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = sqrt(((u * 2.0d0) * (t + (((l * (-2.0d0)) + ((u_42 - u) / ((om / l) / n))) / (om / l))))) * sqrt(n)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -3.6e-279) {
		tmp = Math.sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)));
	} else if (n <= 3.05e-114) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = Math.sqrt(((U * 2.0) * (t + (((l * -2.0) + ((U_42_ - U) / ((Om / l) / n))) / (Om / l))))) * Math.sqrt(n);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -3.6e-279:
		tmp = math.sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)))
	elif n <= 3.05e-114:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = math.sqrt(((U * 2.0) * (t + (((l * -2.0) + ((U_42_ - U) / ((Om / l) / n))) / (Om / l))))) * math.sqrt(n)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -3.6e-279)
		tmp = sqrt(Float64(Float64(Float64(U / Float64(Float64(Om / l) / Float64(Float64(l * -2.0) + Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ - U))))) + Float64(U * t)) * Float64(n * 2.0)));
	elseif (n <= 3.05e-114)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(Float64(Float64(U * 2.0) * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n))) / Float64(Om / l))))) * sqrt(n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -3.6e-279)
		tmp = sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)));
	elseif (n <= 3.05e-114)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = sqrt(((U * 2.0) * (t + (((l * -2.0) + ((U_42_ - U) / ((Om / l) / n))) / (Om / l))))) * sqrt(n);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -3.6e-279], N[Sqrt[N[(N[(N[(U / N[(N[(Om / l), $MachinePrecision] / N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.05e-114], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.5999999999999997e-279

   1. Initial program 48.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 53.5%

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

   4. Applied egg-rr 53.5%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   8. Applied egg-rr 62.1%

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

   if -3.5999999999999997e-279 < n < 3.04999999999999988e-114

   1. Initial program 43.8%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 53.7%

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

   5. Simplified 53.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 71.4%

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

   7. Applied egg-rr 71.4%

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

   if 3.04999999999999988e-114 < n

   1. Initial program 51.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 56.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 64.2%

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

   8. Applied egg-rr 64.2%

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

   10. Applied egg-rr 82.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 3.05 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{t\_1}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 3.05 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \left(t + \frac{\ell}{\frac{Om}{t\_1}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ (* l -2.0) (* (/ n (/ Om l)) (- U* U)))))
   (if (<= n -1.35e-274)
     (sqrt (* (+ (/ U (/ (/ Om l) t_1)) (* U t)) (* n 2.0)))
     (if (<= n 3.05e-114)
       (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
       (* (sqrt n) (sqrt (* (* U 2.0) (+ t (/ l (/ Om t_1))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U));
	double tmp;
	if (n <= -1.35e-274) {
		tmp = sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)));
	} else if (n <= 3.05e-114) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = sqrt(n) * sqrt(((U * 2.0) * (t + (l / (Om / t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * (-2.0d0)) + ((n / (om / l)) * (u_42 - u))
    if (n <= (-1.35d-274)) then
        tmp = sqrt((((u / ((om / l) / t_1)) + (u * t)) * (n * 2.0d0)))
    else if (n <= 3.05d-114) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = sqrt(n) * sqrt(((u * 2.0d0) * (t + (l / (om / t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U));
	double tmp;
	if (n <= -1.35e-274) {
		tmp = Math.sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)));
	} else if (n <= 3.05e-114) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt(((U * 2.0) * (t + (l / (Om / t_1)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U))
	tmp = 0
	if n <= -1.35e-274:
		tmp = math.sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)))
	elif n <= 3.05e-114:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = math.sqrt(n) * math.sqrt(((U * 2.0) * (t + (l / (Om / t_1)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * -2.0) + Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ - U)))
	tmp = 0.0
	if (n <= -1.35e-274)
		tmp = sqrt(Float64(Float64(Float64(U / Float64(Float64(Om / l) / t_1)) + Float64(U * t)) * Float64(n * 2.0)));
	elseif (n <= 3.05e-114)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(U * 2.0) * Float64(t + Float64(l / Float64(Om / t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U));
	tmp = 0.0;
	if (n <= -1.35e-274)
		tmp = sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)));
	elseif (n <= 3.05e-114)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = sqrt(n) * sqrt(((U * 2.0) * (t + (l / (Om / t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-274], N[Sqrt[N[(N[(N[(U / N[(N[(Om / l), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.05e-114], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(t + N[(l / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.35e-274

   1. Initial program 48.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 53.5%

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

   4. Applied egg-rr 53.5%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   8. Applied egg-rr 62.1%

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

   if -1.35e-274 < n < 3.04999999999999988e-114

   1. Initial program 43.8%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 53.7%

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

   5. Simplified 53.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 71.4%

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

   7. Applied egg-rr 71.4%

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

   if 3.04999999999999988e-114 < n

   1. Initial program 51.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 56.3%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   8. Applied egg-rr 81.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 3.05 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(U \cdot 2\right) \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.1 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)}{\frac{Om}{\ell}}\right)} \cdot \sqrt{n \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -6.1e-276)
   (sqrt
    (*
     (+ (/ U (/ (/ Om l) (+ (* l -2.0) (* (/ n (/ Om l)) (- U* U))))) (* U t))
     (* n 2.0)))
   (if (<= n 4.9e-163)
     (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
     (*
      (sqrt
       (* U (+ t (/ (+ (* l -2.0) (* (/ l Om) (* n (- U* U)))) (/ Om l)))))
      (sqrt (* n 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -6.1e-276) {
		tmp = sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)));
	} else if (n <= 4.9e-163) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = sqrt((U * (t + (((l * -2.0) + ((l / Om) * (n * (U_42_ - U)))) / (Om / l))))) * sqrt((n * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-6.1d-276)) then
        tmp = sqrt((((u / ((om / l) / ((l * (-2.0d0)) + ((n / (om / l)) * (u_42 - u))))) + (u * t)) * (n * 2.0d0)))
    else if (n <= 4.9d-163) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = sqrt((u * (t + (((l * (-2.0d0)) + ((l / om) * (n * (u_42 - u)))) / (om / l))))) * sqrt((n * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -6.1e-276) {
		tmp = Math.sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)));
	} else if (n <= 4.9e-163) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = Math.sqrt((U * (t + (((l * -2.0) + ((l / Om) * (n * (U_42_ - U)))) / (Om / l))))) * Math.sqrt((n * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -6.1e-276:
		tmp = math.sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)))
	elif n <= 4.9e-163:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = math.sqrt((U * (t + (((l * -2.0) + ((l / Om) * (n * (U_42_ - U)))) / (Om / l))))) * math.sqrt((n * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -6.1e-276)
		tmp = sqrt(Float64(Float64(Float64(U / Float64(Float64(Om / l) / Float64(Float64(l * -2.0) + Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ - U))))) + Float64(U * t)) * Float64(n * 2.0)));
	elseif (n <= 4.9e-163)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))) / Float64(Om / l))))) * sqrt(Float64(n * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -6.1e-276)
		tmp = sqrt((((U / ((Om / l) / ((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))))) + (U * t)) * (n * 2.0)));
	elseif (n <= 4.9e-163)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = sqrt((U * (t + (((l * -2.0) + ((l / Om) * (n * (U_42_ - U)))) / (Om / l))))) * sqrt((n * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -6.1e-276], N[Sqrt[N[(N[(N[(U / N[(N[(Om / l), $MachinePrecision] / N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.9e-163], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.09999999999999977e-276

   1. Initial program 48.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 53.5%

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

   4. Applied egg-rr 53.5%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   8. Applied egg-rr 62.1%

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

   if -6.09999999999999977e-276 < n < 4.9000000000000003e-163

   1. Initial program 42.5%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 49.8%

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

   5. Simplified 49.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 70.7%

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

   7. Applied egg-rr 70.7%

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

   if 4.9000000000000003e-163 < n

   1. Initial program 51.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.8%

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

   4. Applied egg-rr 52.8%

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

   6. Applied egg-rr 65.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.1 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \frac{\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)}{\frac{Om}{\ell}}\right)} \cdot \sqrt{n \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\\ \mathbf{if}\;n \leq -4.8 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{t\_1}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + t\_1 \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ (* l -2.0) (* (/ n (/ Om l)) (- U* U)))))
   (if (<= n -4.8e-277)
     (sqrt (* (+ (/ U (/ (/ Om l) t_1)) (* U t)) (* n 2.0)))
     (if (<= n 3.5e-162)
       (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
       (sqrt (* (* n 2.0) (* U (+ t (* t_1 (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U));
	double tmp;
	if (n <= -4.8e-277) {
		tmp = sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)));
	} else if (n <= 3.5e-162) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t + (t_1 * (l / Om))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * (-2.0d0)) + ((n / (om / l)) * (u_42 - u))
    if (n <= (-4.8d-277)) then
        tmp = sqrt((((u / ((om / l) / t_1)) + (u * t)) * (n * 2.0d0)))
    else if (n <= 3.5d-162) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t + (t_1 * (l / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U));
	double tmp;
	if (n <= -4.8e-277) {
		tmp = Math.sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)));
	} else if (n <= 3.5e-162) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (t_1 * (l / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U))
	tmp = 0
	if n <= -4.8e-277:
		tmp = math.sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)))
	elif n <= 3.5e-162:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (t_1 * (l / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * -2.0) + Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ - U)))
	tmp = 0.0
	if (n <= -4.8e-277)
		tmp = sqrt(Float64(Float64(Float64(U / Float64(Float64(Om / l) / t_1)) + Float64(U * t)) * Float64(n * 2.0)));
	elseif (n <= 3.5e-162)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(t_1 * Float64(l / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * -2.0) + ((n / (Om / l)) * (U_42_ - U));
	tmp = 0.0;
	if (n <= -4.8e-277)
		tmp = sqrt((((U / ((Om / l) / t_1)) + (U * t)) * (n * 2.0)));
	elseif (n <= 3.5e-162)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = sqrt(((n * 2.0) * (U * (t + (t_1 * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.8e-277], N[Sqrt[N[(N[(N[(U / N[(N[(Om / l), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.5e-162], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(t$95$1 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.8e-277

   1. Initial program 48.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 53.5%

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

   4. Applied egg-rr 53.5%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   8. Applied egg-rr 62.1%

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

   if -4.8e-277 < n < 3.4999999999999999e-162

   1. Initial program 42.5%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 49.8%

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

   5. Simplified 49.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 70.7%

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

   7. Applied egg-rr 70.7%

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

   if 3.4999999999999999e-162 < n

   1. Initial program 51.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.8%

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

   4. Applied egg-rr 52.8%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 65.3%

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

   8. Applied egg-rr 65.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(\frac{U}{\frac{\frac{Om}{\ell}}{\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)}} + U \cdot t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;n \leq -8 \cdot 10^{-285}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* n 2.0)
           (*
            U
            (+ t (* (+ (* l -2.0) (* (/ n (/ Om l)) (- U* U))) (/ l Om))))))))
   (if (<= n -8e-285)
     t_1
     (if (<= n 2.4e-159)
       (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + (((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))) * (l / Om))))));
	double tmp;
	if (n <= -8e-285) {
		tmp = t_1;
	} else if (n <= 2.4e-159) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + (((l * (-2.0d0)) + ((n / (om / l)) * (u_42 - u))) * (l / om))))))
    if (n <= (-8d-285)) then
        tmp = t_1
    else if (n <= 2.4d-159) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + (((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))) * (l / Om))))));
	double tmp;
	if (n <= -8e-285) {
		tmp = t_1;
	} else if (n <= 2.4e-159) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + (((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))) * (l / Om))))))
	tmp = 0
	if n <= -8e-285:
		tmp = t_1
	elif n <= 2.4e-159:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(n / Float64(Om / l)) * Float64(U_42_ - U))) * Float64(l / Om))))))
	tmp = 0.0
	if (n <= -8e-285)
		tmp = t_1;
	elseif (n <= 2.4e-159)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + (((l * -2.0) + ((n / (Om / l)) * (U_42_ - U))) * (l / Om))))));
	tmp = 0.0;
	if (n <= -8e-285)
		tmp = t_1;
	elseif (n <= 2.4e-159)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -8e-285], t$95$1, If[LessEqual[n, 2.4e-159], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.00000000000000059e-285 or 2.39999999999999997e-159 < n

   1. Initial program 49.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 57.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 62.9%

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

   8. Applied egg-rr 62.9%

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

   if -8.00000000000000059e-285 < n < 2.39999999999999997e-159

   1. Initial program 43.0%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 50.8%

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

   5. Simplified 50.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 73.1%

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

   7. Applied egg-rr 73.1%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2 + \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;n \leq -1.28 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* n 2.0)
           (* U (+ t (* (/ l Om) (+ (* l -2.0) (* U* (/ (* n l) Om))))))))))
   (if (<= n -1.28e-284)
     t_1
     (if (<= n 1.55e-161)
       (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))));
	double tmp;
	if (n <= -1.28e-284) {
		tmp = t_1;
	} else if (n <= 1.55e-161) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + ((l / om) * ((l * (-2.0d0)) + (u_42 * ((n * l) / om))))))))
    if (n <= (-1.28d-284)) then
        tmp = t_1
    else if (n <= 1.55d-161) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))));
	double tmp;
	if (n <= -1.28e-284) {
		tmp = t_1;
	} else if (n <= 1.55e-161) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))))
	tmp = 0
	if n <= -1.28e-284:
		tmp = t_1
	elif n <= 1.55e-161:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om))))))))
	tmp = 0.0
	if (n <= -1.28e-284)
		tmp = t_1;
	elseif (n <= 1.55e-161)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))));
	tmp = 0.0;
	if (n <= -1.28e-284)
		tmp = t_1;
	elseif (n <= 1.55e-161)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.28e-284], t$95$1, If[LessEqual[n, 1.55e-161], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.28000000000000006e-284 or 1.5499999999999999e-161 < n

   1. Initial program 49.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 57.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 62.9%

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

   8. Applied egg-rr 62.9%

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

   9. Taylor expanded in U around 0

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

       1. mul-1-neg N/A

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

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

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

       3. associate-/l* N/A

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

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

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

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

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

       6. *-lowering-*.f64 60.9%

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

   11. Simplified 60.9%

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

   if -1.28000000000000006e-284 < n < 1.5499999999999999e-161

   1. Initial program 43.0%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 50.8%

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

   5. Simplified 50.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 73.1%

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

   7. Applied egg-rr 73.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.28 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;n \leq -1.26 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt (* (* n 2.0) (* U (+ t (* (/ l Om) (* U* (/ (* n l) Om)))))))))
   (if (<= n -1.26e-55)
     t_1
     (if (<= n 6.8e-34)
       (sqrt (+ (* (* U 2.0) (* n t)) (/ (* (* (* U l) (* n l)) -4.0) Om)))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	double tmp;
	if (n <= -1.26e-55) {
		tmp = t_1;
	} else if (n <= 6.8e-34) {
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + ((l / om) * (u_42 * ((n * l) / om)))))))
    if (n <= (-1.26d-55)) then
        tmp = t_1
    else if (n <= 6.8d-34) then
        tmp = sqrt((((u * 2.0d0) * (n * t)) + ((((u * l) * (n * l)) * (-4.0d0)) / om)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	double tmp;
	if (n <= -1.26e-55) {
		tmp = t_1;
	} else if (n <= 6.8e-34) {
		tmp = Math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))))
	tmp = 0
	if n <= -1.26e-55:
		tmp = t_1
	elif n <= 6.8e-34:
		tmp = math.sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om)))))))
	tmp = 0.0
	if (n <= -1.26e-55)
		tmp = t_1;
	elseif (n <= 6.8e-34)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * Float64(n * t)) + Float64(Float64(Float64(Float64(U * l) * Float64(n * l)) * -4.0) / Om)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	tmp = 0.0;
	if (n <= -1.26e-55)
		tmp = t_1;
	elseif (n <= 6.8e-34)
		tmp = sqrt((((U * 2.0) * (n * t)) + ((((U * l) * (n * l)) * -4.0) / Om)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.26e-55], t$95$1, If[LessEqual[n, 6.8e-34], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(U * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2599999999999999e-55 or 6.8000000000000001e-34 < n

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in U* around inf

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

       1. associate-/l* N/A

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

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

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

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

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

       4. *-lowering-*.f64 61.5%

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

   11. Simplified 61.5%

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

   if -1.2599999999999999e-55 < n < 6.8000000000000001e-34

   1. Initial program 42.5%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 43.8%

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

   5. Simplified 43.8%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 56.1%

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

   7. Applied egg-rr 56.1%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right) + \frac{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot -4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{if}\;U* \leq -2.1 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt (* (* n 2.0) (* U (+ t (* (/ l Om) (* U* (/ (* n l) Om)))))))))
   (if (<= U* -2.1e+80)
     t_1
     (if (<= U* 1.65e-10)
       (sqrt (* (* n 2.0) (* U (+ t (* (* l -2.0) (/ l Om))))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	double tmp;
	if (U_42_ <= -2.1e+80) {
		tmp = t_1;
	} else if (U_42_ <= 1.65e-10) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + ((l / om) * (u_42 * ((n * l) / om)))))))
    if (u_42 <= (-2.1d+80)) then
        tmp = t_1
    else if (u_42 <= 1.65d-10) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * (-2.0d0)) * (l / om))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	double tmp;
	if (U_42_ <= -2.1e+80) {
		tmp = t_1;
	} else if (U_42_ <= 1.65e-10) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))))
	tmp = 0
	if U_42_ <= -2.1e+80:
		tmp = t_1
	elif U_42_ <= 1.65e-10:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om)))))))
	tmp = 0.0
	if (U_42_ <= -2.1e+80)
		tmp = t_1;
	elseif (U_42_ <= 1.65e-10)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * -2.0) * Float64(l / Om))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	tmp = 0.0;
	if (U_42_ <= -2.1e+80)
		tmp = t_1;
	elseif (U_42_ <= 1.65e-10)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U$42$, -2.1e+80], t$95$1, If[LessEqual[U$42$, 1.65e-10], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.10000000000000001e80 or 1.65e-10 < U*

   1. Initial program 48.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 54.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 60.1%

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

   8. Applied egg-rr 60.1%

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

   9. Taylor expanded in U* around inf

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

       1. associate-/l* N/A

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

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

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

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

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

       4. *-lowering-*.f64 58.0%

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

   11. Simplified 58.0%

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

   if -2.10000000000000001e80 < U* < 1.65e-10

   1. Initial program 48.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.3%

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

   4. Applied egg-rr 52.3%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 58.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 60.8%

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

   8. Applied egg-rr 60.8%

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

   9. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 57.2%

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

   11. Simplified 57.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.1 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -7.9 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \frac{\frac{\ell \cdot \left(n \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{U - U*}}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* n 2.0) (* U (+ t (* (* l -2.0) (/ l Om))))))))
   (if (<= Om -7.9e-75)
     t_1
     (if (<= Om 2.8e-163)
       (sqrt (* (* U -2.0) (/ (/ (* l (* n (* n l))) (/ Om (- U U*))) Om)))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	double tmp;
	if (Om <= -7.9e-75) {
		tmp = t_1;
	} else if (Om <= 2.8e-163) {
		tmp = sqrt(((U * -2.0) * (((l * (n * (n * l))) / (Om / (U - U_42_))) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + ((l * (-2.0d0)) * (l / om))))))
    if (om <= (-7.9d-75)) then
        tmp = t_1
    else if (om <= 2.8d-163) then
        tmp = sqrt(((u * (-2.0d0)) * (((l * (n * (n * l))) / (om / (u - u_42))) / om)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	double tmp;
	if (Om <= -7.9e-75) {
		tmp = t_1;
	} else if (Om <= 2.8e-163) {
		tmp = Math.sqrt(((U * -2.0) * (((l * (n * (n * l))) / (Om / (U - U_42_))) / Om)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))))
	tmp = 0
	if Om <= -7.9e-75:
		tmp = t_1
	elif Om <= 2.8e-163:
		tmp = math.sqrt(((U * -2.0) * (((l * (n * (n * l))) / (Om / (U - U_42_))) / Om)))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * -2.0) * Float64(l / Om))))))
	tmp = 0.0
	if (Om <= -7.9e-75)
		tmp = t_1;
	elseif (Om <= 2.8e-163)
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(Float64(l * Float64(n * Float64(n * l))) / Float64(Om / Float64(U - U_42_))) / Om)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	tmp = 0.0;
	if (Om <= -7.9e-75)
		tmp = t_1;
	elseif (Om <= 2.8e-163)
		tmp = sqrt(((U * -2.0) * (((l * (n * (n * l))) / (Om / (U - U_42_))) / Om)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -7.9e-75], t$95$1, If[LessEqual[Om, 2.8e-163], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(N[(l * N[(n * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -7.9e-75 or 2.8e-163 < Om

   1. Initial program 48.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 52.7%

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

   4. Applied egg-rr 52.7%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 51.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 56.4%

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

   8. Applied egg-rr 56.4%

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

   9. Taylor expanded in n around 0

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

       1. *-lowering-*.f64 52.9%

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

   11. Simplified 52.9%

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

   if -7.9e-75 < Om < 2.8e-163

   1. Initial program 49.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. /-lowering-/.f64 N/A

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

      17. /-lowering-/.f64 N/A

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

      18. --lowering--.f64 42.5%

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

   5. Simplified 42.5%

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

      1. associate-*r/ N/A

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

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

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. associate-*l* N/A

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

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

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

      8. associate-*r* N/A

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

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

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

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

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

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

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

      12. --lowering--.f64 48.5%

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

   7. Applied egg-rr 48.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7.9 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \frac{\frac{\ell \cdot \left(n \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{U - U*}}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -1.1 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* n 2.0) (* U (+ t (* (* l -2.0) (/ l Om))))))))
   (if (<= Om -1.1e-131)
     t_1
     (if (<= Om 1.4e-160)
       (sqrt (* (* U -2.0) (* (* (* n l) (* n l)) (/ (/ (- U U*) Om) Om))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	double tmp;
	if (Om <= -1.1e-131) {
		tmp = t_1;
	} else if (Om <= 1.4e-160) {
		tmp = sqrt(((U * -2.0) * (((n * l) * (n * l)) * (((U - U_42_) / Om) / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + ((l * (-2.0d0)) * (l / om))))))
    if (om <= (-1.1d-131)) then
        tmp = t_1
    else if (om <= 1.4d-160) then
        tmp = sqrt(((u * (-2.0d0)) * (((n * l) * (n * l)) * (((u - u_42) / om) / om))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	double tmp;
	if (Om <= -1.1e-131) {
		tmp = t_1;
	} else if (Om <= 1.4e-160) {
		tmp = Math.sqrt(((U * -2.0) * (((n * l) * (n * l)) * (((U - U_42_) / Om) / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))))
	tmp = 0
	if Om <= -1.1e-131:
		tmp = t_1
	elif Om <= 1.4e-160:
		tmp = math.sqrt(((U * -2.0) * (((n * l) * (n * l)) * (((U - U_42_) / Om) / Om))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * -2.0) * Float64(l / Om))))))
	tmp = 0.0
	if (Om <= -1.1e-131)
		tmp = t_1;
	elseif (Om <= 1.4e-160)
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(Float64(n * l) * Float64(n * l)) * Float64(Float64(Float64(U - U_42_) / Om) / Om))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	tmp = 0.0;
	if (Om <= -1.1e-131)
		tmp = t_1;
	elseif (Om <= 1.4e-160)
		tmp = sqrt(((U * -2.0) * (((n * l) * (n * l)) * (((U - U_42_) / Om) / Om))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -1.1e-131], t$95$1, If[LessEqual[Om, 1.4e-160], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.1e-131 or 1.40000000000000008e-160 < Om

   1. Initial program 47.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 52.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 52.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      16. /-lowering-/.f64 57.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   8. Applied egg-rr 57.1%

      \[\leadsto \sqrt{\left(U \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 51.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   11. Simplified 51.4%

       \[\leadsto \sqrt{\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   if -1.1e-131 < Om < 1.40000000000000008e-160

   1. Initial program 51.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(-2 \cdot U\right) \cdot \frac{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left(\frac{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left(\frac{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left(\frac{\left({\ell}^{2} \cdot {n}^{2}\right) \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left(\left({\ell}^{2} \cdot {n}^{2}\right) \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left({\ell}^{2} \cdot {n}^{2}\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({n}^{2}\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({n}^{2}\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({n}^{2}\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(n \cdot n\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{U - U*}{Om \cdot Om}\right)\right)\right)\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{\frac{U - U*}{Om}}{Om}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\left(\frac{U - U*}{Om}\right), Om\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(U - U*\right), Om\right), Om\right)\right)\right)\right) \]

      18. --lowering--.f64 46.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right), Om\right)\right)\right)\right) \]

   5. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right)}} \]
   6. Step-by-step derivation

      1. unswap-sqr N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right), Om\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(\ell \cdot n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right), Om\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(\ell \cdot n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right), Om\right)\right)\right)\right) \]

      4. *-lowering-*.f64 53.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{*.f64}\left(\ell, n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right), Om\right)\right)\right)\right) \]

   7. Applied egg-rr 53.1%

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right)} \cdot \frac{\frac{U - U*}{Om}}{Om}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.1 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.35e+22)
   (sqrt (* (* n 2.0) (* U (+ t (* (/ l Om) (* U* (/ (* n l) Om)))))))
   (sqrt
    (/
     (* 2.0 (* U (* (* n l) (- (* l -2.0) (/ (* l (* n (- U U*))) Om)))))
     Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.35e+22) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	} else {
		tmp = sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) - ((l * (n * (U - U_42_))) / Om))))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.35d+22) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l / om) * (u_42 * ((n * l) / om)))))))
    else
        tmp = sqrt(((2.0d0 * (u * ((n * l) * ((l * (-2.0d0)) - ((l * (n * (u - u_42))) / om))))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.35e+22) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	} else {
		tmp = Math.sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) - ((l * (n * (U - U_42_))) / Om))))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.35e+22:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))))
	else:
		tmp = math.sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) - ((l * (n * (U - U_42_))) / Om))))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.35e+22)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(l * -2.0) - Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.35e+22)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / Om) * (U_42_ * ((n * l) / Om)))))));
	else
		tmp = sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) - ((l * (n * (U - U_42_))) / Om))))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.35e+22], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.3500000000000001e22

   1. Initial program 53.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 56.2%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      16. /-lowering-/.f64 61.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   8. Applied egg-rr 61.3%

      \[\leadsto \sqrt{\left(U \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

       4. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   11. Simplified 53.6%

       \[\leadsto \sqrt{\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   if 2.3500000000000001e22 < l

   1. Initial program 26.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 38.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 38.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 56.7%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   9. Simplified 60.2%

      \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;Om \leq -8.5 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 1.35 \cdot 10^{-208}:\\ \;\;\;\;\ell \cdot \sqrt{\left(U \cdot -2\right) \cdot \frac{n \cdot n}{\frac{Om}{\frac{U - U*}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* n 2.0) (* U (+ t (* (* l -2.0) (/ l Om))))))))
   (if (<= Om -8.5e-75)
     t_1
     (if (<= Om 1.35e-208)
       (* l (sqrt (* (* U -2.0) (/ (* n n) (/ Om (/ (- U U*) Om))))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	double tmp;
	if (Om <= -8.5e-75) {
		tmp = t_1;
	} else if (Om <= 1.35e-208) {
		tmp = l * sqrt(((U * -2.0) * ((n * n) / (Om / ((U - U_42_) / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * (u * (t + ((l * (-2.0d0)) * (l / om))))))
    if (om <= (-8.5d-75)) then
        tmp = t_1
    else if (om <= 1.35d-208) then
        tmp = l * sqrt(((u * (-2.0d0)) * ((n * n) / (om / ((u - u_42) / om)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	double tmp;
	if (Om <= -8.5e-75) {
		tmp = t_1;
	} else if (Om <= 1.35e-208) {
		tmp = l * Math.sqrt(((U * -2.0) * ((n * n) / (Om / ((U - U_42_) / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))))
	tmp = 0
	if Om <= -8.5e-75:
		tmp = t_1
	elif Om <= 1.35e-208:
		tmp = l * math.sqrt(((U * -2.0) * ((n * n) / (Om / ((U - U_42_) / Om)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * -2.0) * Float64(l / Om))))))
	tmp = 0.0
	if (Om <= -8.5e-75)
		tmp = t_1;
	elseif (Om <= 1.35e-208)
		tmp = Float64(l * sqrt(Float64(Float64(U * -2.0) * Float64(Float64(n * n) / Float64(Om / Float64(Float64(U - U_42_) / Om))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	tmp = 0.0;
	if (Om <= -8.5e-75)
		tmp = t_1;
	elseif (Om <= 1.35e-208)
		tmp = l * sqrt(((U * -2.0) * ((n * n) / (Om / ((U - U_42_) / Om)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -8.5e-75], t$95$1, If[LessEqual[Om, 1.35e-208], N[(l * N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] / N[(Om / N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -8.5000000000000001e-75 or 1.35e-208 < Om

   1. Initial program 47.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 51.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 51.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 51.4%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      16. /-lowering-/.f64 56.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   8. Applied egg-rr 56.3%

      \[\leadsto \sqrt{\left(U \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   11. Simplified 52.4%

       \[\leadsto \sqrt{\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   if -8.5000000000000001e-75 < Om < 1.35e-208

   1. Initial program 53.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(-2 \cdot U\right) \cdot \frac{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left(\frac{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left(\frac{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left(\frac{\left({\ell}^{2} \cdot {n}^{2}\right) \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left(\left({\ell}^{2} \cdot {n}^{2}\right) \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left({\ell}^{2} \cdot {n}^{2}\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({n}^{2}\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({n}^{2}\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({n}^{2}\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(n \cdot n\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{U - U*}{{Om}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{U - U*}{Om \cdot Om}\right)\right)\right)\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{\frac{U - U*}{Om}}{Om}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\left(\frac{U - U*}{Om}\right), Om\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(U - U*\right), Om\right), Om\right)\right)\right)\right) \]

      18. --lowering--.f64 43.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), Om\right), Om\right)\right)\right)\right) \]

   5. Simplified 43.6%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)} \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right)\right) \cdot \left(-2 \cdot U\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\left(\ell \cdot \ell\right) \cdot \left(\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)\right)} \]

      4. sqrt-prod N/A

         \[\leadsto \sqrt{\ell \cdot \ell} \cdot \color{blue}{\sqrt{\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)}} \]

      5. pow2 N/A

         \[\leadsto \sqrt{{\ell}^{2}} \cdot \sqrt{\color{blue}{\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right)} \cdot \left(-2 \cdot U\right)} \]

      6. sqrt-pow1 N/A

         \[\leadsto {\ell}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\color{blue}{\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)}} \]

      7. metadata-eval N/A

         \[\leadsto {\ell}^{1} \cdot \sqrt{\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \color{blue}{\left(-2 \cdot U\right)}} \]

      8. unpow1 N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\sqrt{\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)}\right)}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\left(\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right) \cdot \left(-2 \cdot U\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(n \cdot n\right) \cdot \frac{\frac{U - U*}{Om}}{Om}\right), \left(-2 \cdot U\right)\right)\right)\right) \]

   7. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{\frac{n \cdot n}{\frac{Om}{\frac{U - U*}{Om}}} \cdot \left(U \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.35 \cdot 10^{-208}:\\ \;\;\;\;\ell \cdot \sqrt{\left(U \cdot -2\right) \cdot \frac{n \cdot n}{\frac{Om}{\frac{U - U*}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.8e+213)
   (sqrt (* (* n 2.0) (* U (+ t (* (* l -2.0) (/ l Om))))))
   (sqrt (* (* n 2.0) (/ (* U (* U* (* n (* l l)))) (* Om Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.8e+213) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	} else {
		tmp = sqrt(((n * 2.0) * ((U * (U_42_ * (n * (l * l)))) / (Om * Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.8d+213) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * (-2.0d0)) * (l / om))))))
    else
        tmp = sqrt(((n * 2.0d0) * ((u * (u_42 * (n * (l * l)))) / (om * om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.8e+213) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * ((U * (U_42_ * (n * (l * l)))) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.8e+213:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))))
	else:
		tmp = math.sqrt(((n * 2.0) * ((U * (U_42_ * (n * (l * l)))) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.8e+213)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * -2.0) * Float64(l / Om))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * Float64(U_42_ * Float64(n * Float64(l * l)))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.8e+213)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
	else
		tmp = sqrt(((n * 2.0) * ((U * (U_42_ * (n * (l * l)))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.8e+213], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7999999999999999e213

   1. Initial program 51.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 54.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 54.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 56.2%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      16. /-lowering-/.f64 60.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   8. Applied egg-rr 60.7%

      \[\leadsto \sqrt{\left(U \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 49.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   11. Simplified 49.5%

       \[\leadsto \sqrt{\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

   if 2.7999999999999999e213 < l

   1. Initial program 12.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 17.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 17.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 57.2%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}\right)}, \mathsf{*.f64}\left(2, n\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \left({\ell}^{2} \cdot n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), \left({Om}^{2}\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), \left(Om \cdot Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

      8. *-lowering-*.f64 50.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   9. Simplified 50.6%

      \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left(U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om \cdot Om}} \cdot \left(2 \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{U \cdot \left(U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+170}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -3.3e+170)
   (pow (* n (* (* U t) 2.0)) 0.5)
   (sqrt (* (* U 2.0) (* n (+ t (* -2.0 (/ (* l l) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -3.3e+170) {
		tmp = pow((n * ((U * t) * 2.0)), 0.5);
	} else {
		tmp = sqrt(((U * 2.0) * (n * (t + (-2.0 * ((l * l) / Om))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-3.3d+170)) then
        tmp = (n * ((u * t) * 2.0d0)) ** 0.5d0
    else
        tmp = sqrt(((u * 2.0d0) * (n * (t + ((-2.0d0) * ((l * l) / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -3.3e+170) {
		tmp = Math.pow((n * ((U * t) * 2.0)), 0.5);
	} else {
		tmp = Math.sqrt(((U * 2.0) * (n * (t + (-2.0 * ((l * l) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -3.3e+170:
		tmp = math.pow((n * ((U * t) * 2.0)), 0.5)
	else:
		tmp = math.sqrt(((U * 2.0) * (n * (t + (-2.0 * ((l * l) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -3.3e+170)
		tmp = Float64(n * Float64(Float64(U * t) * 2.0)) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -3.3e+170)
		tmp = (n * ((U * t) * 2.0)) ^ 0.5;
	else
		tmp = sqrt(((U * 2.0) * (n * (t + (-2.0 * ((l * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -3.3e+170], N[Power[N[(n * N[(N[(U * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.30000000000000023e170

   1. Initial program 39.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 48.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 48.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(n \cdot U\right), t\right)\right)\right) \]

      5. *-lowering-*.f64 48.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, U\right), t\right)\right)\right) \]

   7. Simplified 48.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot 2\right), \frac{1}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(\left(U \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(U \cdot t\right), 2\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 51.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, t\right), 2\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 51.1%

      \[\leadsto \color{blue}{{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}} \]

   if -3.30000000000000023e170 < t

   1. Initial program 50.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   4. 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. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. metadata-eval 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. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(\frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{Om}\right), -2\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right), -2\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right), -2\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 44.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), -2\right)\right)\right)\right)\right) \]

   5. Simplified 44.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+170}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+83}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.12e+83)
   (pow (* n (* (* U t) 2.0)) 0.5)
   (sqrt (/ (* -4.0 (* n (* U (* l l)))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.12e+83) {
		tmp = pow((n * ((U * t) * 2.0)), 0.5);
	} else {
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.12d+83) then
        tmp = (n * ((u * t) * 2.0d0)) ** 0.5d0
    else
        tmp = sqrt((((-4.0d0) * (n * (u * (l * l)))) / om))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.12e+83) {
		tmp = Math.pow((n * ((U * t) * 2.0)), 0.5);
	} else {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.12e+83:
		tmp = math.pow((n * ((U * t) * 2.0)), 0.5)
	else:
		tmp = math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.12e+83)
		tmp = Float64(n * Float64(Float64(U * t) * 2.0)) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l * l)))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.12e+83)
		tmp = (n * ((U * t) * 2.0)) ^ 0.5;
	else
		tmp = sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.12e+83], N[Power[N[(n * N[(N[(U * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.12e83

   1. Initial program 54.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(n \cdot U\right), t\right)\right)\right) \]

      5. *-lowering-*.f64 39.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, U\right), t\right)\right)\right) \]

   7. Simplified 39.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot 2\right), \frac{1}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(\left(U \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(U \cdot t\right), 2\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 40.9%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, t\right), 2\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}} \]

   if 1.12e83 < l

   1. Initial program 18.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4\right), Om\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), -4\right), Om\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), -4\right), Om\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), -4\right), Om\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), -4\right), Om\right)\right)\right) \]

      14. *-lowering-*.f64 22.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), -4\right), Om\right)\right)\right) \]

   5. Simplified 22.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   7. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\left(U \cdot {\ell}^{2}\right) \cdot n\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\left(U \cdot {\ell}^{2}\right), n\right)\right), Om\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2}\right)\right), n\right)\right), Om\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(\ell \cdot \ell\right)\right), n\right)\right), Om\right)\right) \]

      8. *-lowering-*.f64 22.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\ell, \ell\right)\right), n\right)\right), Om\right)\right) \]

   8. Simplified 22.3%

      \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+83}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* (* n 2.0) (* U (+ t (* (* l -2.0) (/ l Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
}

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(((n * 2.0d0) * (u * (t + ((l * (-2.0d0)) * (l / om))))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * -2.0) * Float64(l / Om))))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((n * 2.0) * (U * (t + ((l * -2.0) * (l / Om))))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 52.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

4. Applied egg-rr 52.0%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
5. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{1}{\frac{Om}{\ell}}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right), \left(2 \cdot n\right)\right)\right) \]

6. Applied egg-rr 56.3%

   \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(t + \frac{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)}}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   14. un-div-inv N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

   16. /-lowering-/.f64 60.5%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

8. Applied egg-rr 60.5%

   \[\leadsto \sqrt{\left(U \cdot \left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

9. Taylor expanded in n around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 47.7%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(2, n\right)\right)\right) \]

11. Simplified 47.7%

    \[\leadsto \sqrt{\left(U \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)\right) \cdot \left(2 \cdot n\right)} \]

12. Final simplification 47.7%

    \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\ell \cdot -2\right) \cdot \frac{\ell}{Om}\right)\right)} \]
13. Add Preprocessing

Alternative 17: 36.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -8.8e-269)
   (sqrt (* n (* t (* U 2.0))))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -8.8e-269) {
		tmp = sqrt((n * (t * (U * 2.0))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-8.8d-269)) then
        tmp = sqrt((n * (t * (u * 2.0d0))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -8.8e-269) {
		tmp = Math.sqrt((n * (t * (U * 2.0))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -8.8e-269:
		tmp = math.sqrt((n * (t * (U * 2.0))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -8.8e-269)
		tmp = sqrt(Float64(n * Float64(t * Float64(U * 2.0))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -8.8e-269)
		tmp = sqrt((n * (t * (U * 2.0))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -8.8e-269], N[Sqrt[N[(n * N[(t * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.79999999999999936e-269

   1. Initial program 48.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. 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 27.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   5. Simplified 27.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. 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 36.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), t\right), n\right)\right) \]

   7. Applied egg-rr 36.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}} \]

   if -8.79999999999999936e-269 < n

   1. Initial program 48.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. 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 38.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   5. Simplified 38.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. 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. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-commutative 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 39.2%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, n\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 39.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.32e-269)
   (sqrt (* 2.0 (* t (* n U))))
   (sqrt (* (* U 2.0) (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.32e-269) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt(((U * 2.0) * (n * t)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-1.32d-269)) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt(((u * 2.0d0) * (n * t)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.32e-269) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt(((U * 2.0) * (n * t)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -1.32e-269:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt(((U * 2.0) * (n * t)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.32e-269)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(U * 2.0) * Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -1.32e-269)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt(((U * 2.0) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.32e-269], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * 2.0), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.32000000000000007e-269

   1. Initial program 48.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 53.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 53.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(n \cdot U\right), t\right)\right)\right) \]

      5. *-lowering-*.f64 34.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, U\right), t\right)\right)\right) \]

   7. Simplified 34.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

   if -1.32000000000000007e-269 < n

   1. Initial program 48.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. 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 38.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   5. Simplified 38.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* n (* (* U t) 2.0)) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((n * ((U * t) * 2.0)), 0.5);
}

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 = (n * ((u * t) * 2.0d0)) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((n * ((U * t) * 2.0)), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((n * ((U * t) * 2.0)), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(n * Float64(Float64(U * t) * 2.0)) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (n * ((U * t) * 2.0)) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(n * N[(N[(U * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 52.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

4. Applied egg-rr 52.0%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(n \cdot U\right), t\right)\right)\right) \]

   5. *-lowering-*.f64 34.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, U\right), t\right)\right)\right) \]

7. Simplified 34.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. pow1/2 N/A

      \[\leadsto {\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right), \frac{1}{2}\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot 2\right), \frac{1}{2}\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(\left(U \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(U \cdot t\right), 2\right)\right), \frac{1}{2}\right) \]

   8. *-lowering-*.f64 36.4%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, t\right), 2\right)\right), \frac{1}{2}\right) \]

9. Applied egg-rr 36.4%

   \[\leadsto \color{blue}{{\left(n \cdot \left(\left(U \cdot t\right) \cdot 2\right)\right)}^{0.5}} \]
10. Add Preprocessing

Alternative 20: 34.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* n (* t (* U 2.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((n * (t * (U * 2.0))));
}

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((n * (t * (u * 2.0d0))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((n * (t * (U * 2.0))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((n * (t * (U * 2.0))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(n * Float64(t * Float64(U * 2.0))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((n * (t * (U * 2.0))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(n * N[(t * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
4. 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 33.2%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

5. Simplified 33.2%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
6. 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 36.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), t\right), n\right)\right) \]

7. Applied egg-rr 36.0%

   \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}} \]

8. Final simplification 36.0%

   \[\leadsto \sqrt{n \cdot \left(t \cdot \left(U \cdot 2\right)\right)} \]
9. Add Preprocessing

Alternative 21: 35.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* t (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (t * (n * U))));
}

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 * (t * (n * u))))
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 * (t * (n * U))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (t * (n * U))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(t * Float64(n * U))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (t * (n * U))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 52.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

4. Applied egg-rr 52.0%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot n\right) \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot n\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(n \cdot U\right), t\right)\right)\right) \]

   5. *-lowering-*.f64 34.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, U\right), t\right)\right)\right) \]

7. Simplified 34.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

8. Final simplification 34.6%

   \[\leadsto \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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*))))))