Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 64.9%

Time: 29.4s

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: 49.8% 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: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;n \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t_1\right)}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* (/ l Om) (fma l -2.0 (* (* n (/ l Om)) (- U* U)))))))
   (if (<= n -1.45e-117)
     (sqrt (* 2.0 (* (* n U) t_1)))
     (if (<= n -1.5e-230)
       (sqrt
        (*
         (* n 2.0)
         (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
       (if (<= n -1e-309)
         (sqrt
          (*
           2.0
           (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ (* U (* n l)) Om)))))))))
         (* (sqrt (* n 2.0)) (sqrt (* U t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + ((l / Om) * fma(l, -2.0, ((n * (l / Om)) * (U_42_ - U))));
	double tmp;
	if (n <= -1.45e-117) {
		tmp = sqrt((2.0 * ((n * U) * t_1)));
	} else if (n <= -1.5e-230) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U)))))))));
	} else if (n <= -1e-309) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(l / Om)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (n <= -1.45e-117)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * t_1)));
	elseif (n <= -1.5e-230)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	elseif (n <= -1e-309)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(Float64(U * Float64(n * l)) / Om)))))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * t_1)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.45e-117], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -1.5e-230], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -1e-309], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.45e-117

   1. Initial program 55.2%

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

      1. associate-*l* 53.6%

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

      2. sub-neg 53.6%

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

      3. associate--l+ 53.6%

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

      4. *-commutative 53.6%

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

      5. distribute-rgt-neg-in 53.6%

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

      6. associate-*l/ 53.7%

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

      7. associate-*l* 53.7%

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

      8. *-commutative 53.7%

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

      9. *-commutative 53.7%

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

      10. associate-*l* 47.6%

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

      11. unpow2 47.6%

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

      12. associate-*l* 47.9%

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

   3. Simplified 51.5%

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

      1. *-un-lft-identity 51.5%

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

      2. associate-*l* 51.5%

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

   5. Applied egg-rr 51.5%

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

      1. *-lft-identity 51.5%

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

      2. associate-*r* 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-*r* 59.8%

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

   7. Simplified 59.8%

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

   if -1.45e-117 < n < -1.5e-230

   1. Initial program 39.0%

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

      1. associate-*l* 60.9%

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

      2. sub-neg 60.9%

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

      3. associate--l+ 60.9%

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

      4. *-commutative 60.9%

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

      5. distribute-rgt-neg-in 60.9%

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

      6. associate-*l/ 69.8%

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

      7. associate-*l* 69.8%

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

      8. *-commutative 69.8%

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

      9. *-commutative 69.8%

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

      10. associate-*l* 69.7%

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

      11. unpow2 69.7%

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

      12. associate-*l* 69.7%

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

   3. Simplified 69.7%

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

   if -1.5e-230 < n < -1.000000000000002e-309

   1. Initial program 27.4%

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

      1. associate-*l* 21.7%

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

      2. sub-neg 21.7%

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

      3. associate--l+ 21.7%

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

      4. *-commutative 21.7%

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

      5. distribute-rgt-neg-in 21.7%

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

      6. associate-*l/ 40.1%

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

      7. associate-*l* 40.1%

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

      8. *-commutative 40.1%

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

      9. *-commutative 40.1%

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

      10. associate-*l* 40.1%

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

      11. unpow2 40.1%

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

      12. associate-*l* 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in U* around 0 21.2%

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

      1. associate-*r* 27.3%

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

      2. +-commutative 27.3%

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

   6. Simplified 64.4%

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

   if -1.000000000000002e-309 < n

   1. Initial program 55.4%

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

      1. associate-*l* 53.0%

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

      2. sub-neg 53.0%

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

      3. associate--l+ 53.0%

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

      4. *-commutative 53.0%

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

      5. distribute-rgt-neg-in 53.0%

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

      6. associate-*l/ 56.1%

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

      7. associate-*l* 56.1%

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

      8. *-commutative 56.1%

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

      9. *-commutative 56.1%

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

      10. associate-*l* 52.1%

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

      11. unpow2 52.1%

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

      12. associate-*l* 54.5%

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

   3. Simplified 58.5%

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

      1. sqrt-prod 66.9%

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

      2. *-commutative 66.9%

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

   5. Applied egg-rr 66.9%

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

      1. *-commutative 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-*r* 72.3%

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

   7. Simplified 72.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \]

Alternative 2: 65.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t + \left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\frac{n \cdot U*}{Om} - 2}{Om}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* U (* n 2.0))
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (- U* U) (* n (pow (/ l Om) 2.0))))))))
   (if (<= t_1 0.0)
     (*
      (sqrt (* n 2.0))
      (sqrt (+ (* U t) (* (* l (* U l)) (/ (- (/ (* n U*) Om) 2.0) Om)))))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         2.0
         (*
          (* n U)
          (+ t (* (/ l Om) (fma l -2.0 (* (* n (/ l Om)) (- U* U))))))))
       (sqrt
        (*
         2.0
         (+
          (/ (+ (* l -2.0) (/ (* (- U* U) (* n l)) Om)) (/ Om (* n (* U l))))
          (* n (* U t)))))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 8.9%

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

      1. associate-*l* 28.0%

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

      2. sub-neg 28.0%

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

      3. associate--l+ 28.0%

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

      4. *-commutative 28.0%

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

      5. distribute-rgt-neg-in 28.0%

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

      6. associate-*l/ 28.0%

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

      7. associate-*l* 28.0%

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

      8. *-commutative 28.0%

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

      9. *-commutative 28.0%

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

      10. associate-*l* 27.9%

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

      11. unpow2 27.9%

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

      12. associate-*l* 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in l around -inf 24.8%

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

      1. mul-1-neg 24.8%

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

      2. unsub-neg 24.8%

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

      3. *-commutative 24.8%

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

      4. associate-/l* 24.8%

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

      5. mul-1-neg 24.8%

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

      6. unsub-neg 24.8%

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

      7. associate-/l* 24.6%

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

      8. *-commutative 24.6%

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

      9. unpow2 24.6%

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

   6. Simplified 24.6%

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

      1. sqrt-prod 33.5%

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

      2. associate-/r/ 33.6%

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

      3. associate-/r/ 33.6%

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

      4. associate-*r* 33.6%

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

   8. Applied egg-rr 33.6%

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

   9. Taylor expanded in U* around inf 33.6%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 68.2%

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

      1. associate-*l* 64.7%

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

      2. sub-neg 64.7%

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

      3. associate--l+ 64.7%

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

      4. *-commutative 64.7%

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

      5. distribute-rgt-neg-in 64.7%

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

      6. associate-*l/ 67.2%

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

      7. associate-*l* 67.2%

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

      8. *-commutative 67.2%

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

      9. *-commutative 67.2%

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

      10. associate-*l* 61.8%

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

      11. unpow2 61.8%

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

      12. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. associate-*l* 61.9%

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

   5. Applied egg-rr 61.9%

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

      1. *-lft-identity 61.9%

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

      2. associate-*r* 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r* 72.2%

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

   7. Simplified 72.2%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 0.0%

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

      1. associate-*l* 1.1%

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

      2. sub-neg 1.1%

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

      3. associate--l+ 1.1%

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

      4. *-commutative 1.1%

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

      5. distribute-rgt-neg-in 1.1%

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

      6. associate-*l/ 12.9%

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

      7. associate-*l* 12.9%

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

      8. *-commutative 12.9%

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

      9. *-commutative 12.9%

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

      10. associate-*l* 12.9%

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

      11. unpow2 12.9%

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

      12. associate-*l* 22.0%

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

   3. Simplified 46.7%

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

      1. *-un-lft-identity 46.7%

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

      2. associate-*l* 46.7%

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

   5. Applied egg-rr 46.7%

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

      1. *-lft-identity 46.7%

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

      2. associate-*r* 32.3%

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

      3. *-commutative 32.3%

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

      4. associate-*r* 34.7%

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

   7. Simplified 34.7%

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

   8. Taylor expanded in t around 0 48.8%

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

      1. associate-/l* 48.9%

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

      2. +-commutative 48.9%

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

      3. *-commutative 48.9%

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

      4. associate-*r* 54.4%

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

      5. *-commutative 54.4%

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

   10. Simplified 54.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t + \left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\frac{n \cdot U*}{Om} - 2}{Om}}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 3: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -3.3 \cdot 10^{-141} \lor \neg \left(U \leq 1.05 \cdot 10^{-150}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U -3.3e-141) (not (<= U 1.05e-150)))
   (sqrt
    (*
     2.0
     (* (* n U) (+ t (* (/ l Om) (fma l -2.0 (* (* n (/ l Om)) (- U* U))))))))
   (sqrt
    (-
     (* 2.0 (* n (* U t)))
     (*
      2.0
      (/ (* (* n (* U l)) (- (/ (* n (* l (- U U*))) Om) (* l -2.0))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -3.3e-141) || !(U <= 1.05e-150)) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l / Om) * fma(l, -2.0, ((n * (l / Om)) * (U_42_ - U))))))));
	} else {
		tmp = sqrt(((2.0 * (n * (U * t))) - (2.0 * (((n * (U * l)) * (((n * (l * (U - U_42_))) / Om) - (l * -2.0))) / Om))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U <= -3.3e-141) || !(U <= 1.05e-150))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(l / Om)) * Float64(U_42_ - U))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) - Float64(2.0 * Float64(Float64(Float64(n * Float64(U * l)) * Float64(Float64(Float64(n * Float64(l * Float64(U - U_42_))) / Om) - Float64(l * -2.0))) / Om))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U, -3.3e-141], N[Not[LessEqual[U, 1.05e-150]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[(l * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -3.29999999999999999e-141 or 1.0500000000000001e-150 < U

   1. Initial program 60.1%

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

      1. associate-*l* 57.9%

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

      2. sub-neg 57.9%

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

      3. associate--l+ 57.9%

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

      4. *-commutative 57.9%

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

      5. distribute-rgt-neg-in 57.9%

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

      6. associate-*l/ 62.5%

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

      7. associate-*l* 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. associate-*l* 58.7%

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

      11. unpow2 58.7%

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

      12. associate-*l* 58.9%

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

   3. Simplified 61.4%

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

      1. *-un-lft-identity 61.4%

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

      2. associate-*l* 61.4%

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

   5. Applied egg-rr 61.4%

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

      1. *-lft-identity 61.4%

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

      2. associate-*r* 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r* 69.1%

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

   7. Simplified 69.1%

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

   if -3.29999999999999999e-141 < U < 1.0500000000000001e-150

   1. Initial program 36.2%

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

      1. associate-*l* 39.9%

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

      2. sub-neg 39.9%

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

      3. associate--l+ 39.9%

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

      4. *-commutative 39.9%

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

      5. distribute-rgt-neg-in 39.9%

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

      6. associate-*l/ 41.1%

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

      7. associate-*l* 41.1%

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

      8. *-commutative 41.1%

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

      9. *-commutative 41.1%

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

      10. associate-*l* 36.4%

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

      11. unpow2 36.4%

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

      12. associate-*l* 40.1%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in t around inf 55.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -3.3 \cdot 10^{-141} \lor \neg \left(U \leq 1.05 \cdot 10^{-150}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \end{array} \]

Alternative 4: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 - \left(U - U*\right) \cdot \frac{n}{Om}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1e+106)
   (sqrt
    (*
     2.0
     (+
      (/ (+ (* l -2.0) (/ (* (- U* U) (* n l)) Om)) (/ Om (* n (* U l))))
      (* n (* U t)))))
   (if (<= l 1.1e+72)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (*
      (* l (sqrt 2.0))
      (sqrt (/ n (/ Om (* U (- -2.0 (* (- U U*) (/ n Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1e+106) {
		tmp = sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	} else if (l <= 1.1e+72) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 - ((U - U_42_) * (n / 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 <= (-1d+106)) then
        tmp = sqrt((2.0d0 * ((((l * (-2.0d0)) + (((u_42 - u) * (n * l)) / om)) / (om / (n * (u * l)))) + (n * (u * t)))))
    else if (l <= 1.1d+72) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n / (om / (u * ((-2.0d0) - ((u - u_42) * (n / 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 <= -1e+106) {
		tmp = Math.sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	} else if (l <= 1.1e+72) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n / (Om / (U * (-2.0 - ((U - U_42_) * (n / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -1e+106:
		tmp = math.sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))))
	elif l <= 1.1e+72:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n / (Om / (U * (-2.0 - ((U - U_42_) * (n / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1e+106)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(l * -2.0) + Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om)) / Float64(Om / Float64(n * Float64(U * l)))) + Float64(n * Float64(U * t)))));
	elseif (l <= 1.1e+72)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 - Float64(Float64(U - U_42_) * Float64(n / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -1e+106)
		tmp = sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	elseif (l <= 1.1e+72)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 - ((U - U_42_) * (n / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1e+106], N[Sqrt[N[(2.0 * N[(N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.1e+72], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 - N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.00000000000000009e106

   1. Initial program 30.0%

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

      1. associate-*l* 27.4%

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

      2. sub-neg 27.4%

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

      3. associate--l+ 27.4%

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

      4. *-commutative 27.4%

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

      5. distribute-rgt-neg-in 27.4%

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

      6. associate-*l/ 47.2%

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

      7. associate-*l* 47.2%

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

      8. *-commutative 47.2%

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

      9. *-commutative 47.2%

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

      10. associate-*l* 50.0%

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

      11. unpow2 50.0%

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

      12. associate-*l* 50.1%

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

   3. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. associate-*l* 58.8%

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

   5. Applied egg-rr 58.8%

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

      1. *-lft-identity 58.8%

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

      2. associate-*r* 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-*r* 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in t around 0 50.1%

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

      1. associate-/l* 53.0%

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

      2. +-commutative 53.0%

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

      3. *-commutative 53.0%

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

      4. associate-*r* 61.6%

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

      5. *-commutative 61.6%

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

   10. Simplified 61.6%

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

   if -1.00000000000000009e106 < l < 1.1e72

   1. Initial program 62.0%

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

      1. associate-*l* 62.7%

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

      2. sub-neg 62.7%

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

      3. associate--l+ 62.7%

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

      4. *-commutative 62.7%

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

      5. distribute-rgt-neg-in 62.7%

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

      6. associate-*l/ 62.7%

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

      7. associate-*l* 62.7%

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

      8. *-commutative 62.7%

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

      9. *-commutative 62.7%

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

      10. associate-*l* 56.9%

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

      11. unpow2 56.9%

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

      12. associate-*l* 58.7%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in U around 0 64.6%

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

   if 1.1e72 < l

   1. Initial program 31.0%

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

      1. associate-*l* 29.1%

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

      2. sub-neg 29.1%

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

      3. associate--l+ 29.1%

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

      4. *-commutative 29.1%

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

      5. distribute-rgt-neg-in 29.1%

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

      6. associate-*l/ 33.4%

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

      7. associate-*l* 33.4%

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

      8. *-commutative 33.4%

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

      9. *-commutative 33.4%

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

      10. associate-*l* 30.6%

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

      11. unpow2 30.6%

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

      12. associate-*l* 30.9%

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

   3. Simplified 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. associate-*l* 40.6%

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

   5. Applied egg-rr 40.6%

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

      1. *-lft-identity 40.6%

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

      2. associate-*r* 46.5%

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

      3. *-commutative 46.5%

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

      4. associate-*r* 46.7%

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

   7. Simplified 46.7%

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

   8. Taylor expanded in t around 0 35.3%

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

      1. associate-/l* 35.2%

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

      2. +-commutative 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-*r* 37.7%

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

      5. *-commutative 37.7%

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

   10. Simplified 37.7%

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

   11. Taylor expanded in l around inf 52.4%

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

       1. associate-/l* 54.8%

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

       2. *-commutative 54.8%

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

       3. sub-neg 54.8%

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

       4. *-commutative 54.8%

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

       5. associate-*r/ 54.9%

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

       6. metadata-eval 54.9%

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

   13. Simplified 54.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 - \left(U - U*\right) \cdot \frac{n}{Om}\right)}}}\\ \end{array} \]

Alternative 5: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -5.1e+104)
   (sqrt
    (*
     2.0
     (+
      (/ (+ (* l -2.0) (/ (* (- U* U) (* n l)) Om)) (/ Om (* n (* U l))))
      (* n (* U t)))))
   (if (<= l 1.5e+73)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (* (* l (sqrt 2.0)) (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om U*)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -5.1e+104) {
		tmp = sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	} else if (l <= 1.5e+73) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 + (n / (Om / U_42_)))))));
	}
	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 <= (-5.1d+104)) then
        tmp = sqrt((2.0d0 * ((((l * (-2.0d0)) + (((u_42 - u) * (n * l)) / om)) / (om / (n * (u * l)))) + (n * (u * t)))))
    else if (l <= 1.5d+73) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n / (om / (u * ((-2.0d0) + (n / (om / u_42)))))))
    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 <= -5.1e+104) {
		tmp = Math.sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	} else if (l <= 1.5e+73) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n / (Om / (U * (-2.0 + (n / (Om / U_42_)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -5.1e+104:
		tmp = math.sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))))
	elif l <= 1.5e+73:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n / (Om / (U * (-2.0 + (n / (Om / U_42_)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -5.1e+104)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(l * -2.0) + Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om)) / Float64(Om / Float64(n * Float64(U * l)))) + Float64(n * Float64(U * t)))));
	elseif (l <= 1.5e+73)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / U_42_))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -5.1e+104)
		tmp = sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	elseif (l <= 1.5e+73)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 + (n / (Om / U_42_)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -5.1e+104], N[Sqrt[N[(2.0 * N[(N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.5e+73], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.1000000000000002e104

   1. Initial program 30.0%

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

      1. associate-*l* 27.4%

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

      2. sub-neg 27.4%

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

      3. associate--l+ 27.4%

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

      4. *-commutative 27.4%

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

      5. distribute-rgt-neg-in 27.4%

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

      6. associate-*l/ 47.2%

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

      7. associate-*l* 47.2%

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

      8. *-commutative 47.2%

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

      9. *-commutative 47.2%

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

      10. associate-*l* 50.0%

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

      11. unpow2 50.0%

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

      12. associate-*l* 50.1%

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

   3. Simplified 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. associate-*l* 58.8%

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

   5. Applied egg-rr 58.8%

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

      1. *-lft-identity 58.8%

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

      2. associate-*r* 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-*r* 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in t around 0 50.1%

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

      1. associate-/l* 53.0%

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

      2. +-commutative 53.0%

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

      3. *-commutative 53.0%

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

      4. associate-*r* 61.6%

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

      5. *-commutative 61.6%

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

   10. Simplified 61.6%

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

   if -5.1000000000000002e104 < l < 1.50000000000000005e73

   1. Initial program 62.0%

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

      1. associate-*l* 62.7%

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

      2. sub-neg 62.7%

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

      3. associate--l+ 62.7%

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

      4. *-commutative 62.7%

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

      5. distribute-rgt-neg-in 62.7%

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

      6. associate-*l/ 62.7%

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

      7. associate-*l* 62.7%

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

      8. *-commutative 62.7%

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

      9. *-commutative 62.7%

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

      10. associate-*l* 56.9%

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

      11. unpow2 56.9%

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

      12. associate-*l* 58.7%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in U around 0 64.6%

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

   if 1.50000000000000005e73 < l

   1. Initial program 31.0%

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

      1. associate-*l* 29.1%

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

      2. sub-neg 29.1%

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

      3. associate--l+ 29.1%

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

      4. *-commutative 29.1%

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

      5. distribute-rgt-neg-in 29.1%

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

      6. associate-*l/ 33.4%

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

      7. associate-*l* 33.4%

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

      8. *-commutative 33.4%

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

      9. *-commutative 33.4%

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

      10. associate-*l* 30.6%

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

      11. unpow2 30.6%

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

      12. associate-*l* 30.9%

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

   3. Simplified 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. associate-*l* 40.6%

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

   5. Applied egg-rr 40.6%

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

      1. *-lft-identity 40.6%

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

      2. associate-*r* 46.5%

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

      3. *-commutative 46.5%

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

      4. associate-*r* 46.7%

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

   7. Simplified 46.7%

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

   8. Taylor expanded in t around 0 35.3%

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

      1. associate-/l* 35.2%

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

      2. +-commutative 35.2%

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

      3. *-commutative 35.2%

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

      4. associate-*r* 37.7%

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

      5. *-commutative 37.7%

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

   10. Simplified 37.7%

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

   11. Taylor expanded in U* around inf 36.3%

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

   12. Taylor expanded in l around inf 51.7%

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

       1. *-commutative 51.7%

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

       2. *-commutative 51.7%

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

       3. associate-/l* 54.0%

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

       4. *-commutative 54.0%

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

       5. sub-neg 54.0%

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

       6. associate-/l* 54.0%

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

       7. metadata-eval 54.0%

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

       8. *-commutative 54.0%

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

   14. Simplified 54.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.1 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 6: 46.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-135}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* n 2.0)
           (/ (* U (* l l)) (/ Om (- -2.0 (* (- U U*) (/ n Om))))))))
        (t_2 (sqrt (* (* n 2.0) (- (* U t) (* 2.0 (/ (* l l) (/ Om U))))))))
   (if (<= l -2.7e-17)
     t_1
     (if (<= l -2.1e-88)
       t_2
       (if (<= l -1.1e-106)
         (sqrt (* n (/ (* 2.0 (* n (* (* l l) (* U U*)))) (* Om Om))))
         (if (<= l 2.05e-135)
           (pow (* 2.0 (* n (* U t))) 0.5)
           (if (<= l 4.2e-40)
             (sqrt (* 2.0 (* (* n U) (+ t (* -2.0 (/ (* l l) Om))))))
             (if (<= l 1.7e+45) t_2 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 * (l * l)) / (Om / (-2.0 - ((U - U_42_) * (n / Om)))))));
	double t_2 = sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	double tmp;
	if (l <= -2.7e-17) {
		tmp = t_1;
	} else if (l <= -2.1e-88) {
		tmp = t_2;
	} else if (l <= -1.1e-106) {
		tmp = sqrt((n * ((2.0 * (n * ((l * l) * (U * U_42_)))) / (Om * Om))));
	} else if (l <= 2.05e-135) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 4.2e-40) {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else if (l <= 1.7e+45) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((n * 2.0d0) * ((u * (l * l)) / (om / ((-2.0d0) - ((u - u_42) * (n / om)))))))
    t_2 = sqrt(((n * 2.0d0) * ((u * t) - (2.0d0 * ((l * l) / (om / u))))))
    if (l <= (-2.7d-17)) then
        tmp = t_1
    else if (l <= (-2.1d-88)) then
        tmp = t_2
    else if (l <= (-1.1d-106)) then
        tmp = sqrt((n * ((2.0d0 * (n * ((l * l) * (u * u_42)))) / (om * om))))
    else if (l <= 2.05d-135) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 4.2d-40) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l * l) / om))))))
    else if (l <= 1.7d+45) then
        tmp = t_2
    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 * (l * l)) / (Om / (-2.0 - ((U - U_42_) * (n / Om)))))));
	double t_2 = Math.sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	double tmp;
	if (l <= -2.7e-17) {
		tmp = t_1;
	} else if (l <= -2.1e-88) {
		tmp = t_2;
	} else if (l <= -1.1e-106) {
		tmp = Math.sqrt((n * ((2.0 * (n * ((l * l) * (U * U_42_)))) / (Om * Om))));
	} else if (l <= 2.05e-135) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 4.2e-40) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else if (l <= 1.7e+45) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * ((U * (l * l)) / (Om / (-2.0 - ((U - U_42_) * (n / Om)))))))
	t_2 = math.sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))))
	tmp = 0
	if l <= -2.7e-17:
		tmp = t_1
	elif l <= -2.1e-88:
		tmp = t_2
	elif l <= -1.1e-106:
		tmp = math.sqrt((n * ((2.0 * (n * ((l * l) * (U * U_42_)))) / (Om * Om))))
	elif l <= 2.05e-135:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 4.2e-40:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	elif l <= 1.7e+45:
		tmp = t_2
	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(Float64(U * Float64(l * l)) / Float64(Om / Float64(-2.0 - Float64(Float64(U - U_42_) * Float64(n / Om)))))))
	t_2 = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) - Float64(2.0 * Float64(Float64(l * l) / Float64(Om / U))))))
	tmp = 0.0
	if (l <= -2.7e-17)
		tmp = t_1;
	elseif (l <= -2.1e-88)
		tmp = t_2;
	elseif (l <= -1.1e-106)
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(n * Float64(Float64(l * l) * Float64(U * U_42_)))) / Float64(Om * Om))));
	elseif (l <= 2.05e-135)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 4.2e-40)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	elseif (l <= 1.7e+45)
		tmp = t_2;
	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 * (l * l)) / (Om / (-2.0 - ((U - U_42_) * (n / Om)))))));
	t_2 = sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	tmp = 0.0;
	if (l <= -2.7e-17)
		tmp = t_1;
	elseif (l <= -2.1e-88)
		tmp = t_2;
	elseif (l <= -1.1e-106)
		tmp = sqrt((n * ((2.0 * (n * ((l * l) * (U * U_42_)))) / (Om * Om))));
	elseif (l <= 2.05e-135)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 4.2e-40)
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	elseif (l <= 1.7e+45)
		tmp = t_2;
	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[(N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(-2.0 - N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] - N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.7e-17], t$95$1, If[LessEqual[l, -2.1e-88], t$95$2, If[LessEqual[l, -1.1e-106], N[Sqrt[N[(n * N[(N[(2.0 * N[(n * N[(N[(l * l), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.05e-135], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 4.2e-40], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.7e+45], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -2.7000000000000001e-17 or 1.7e45 < l

   1. Initial program 36.6%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. associate--l+ 34.9%

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

      4. *-commutative 34.9%

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

      5. distribute-rgt-neg-in 34.9%

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

      6. associate-*l/ 43.2%

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

      7. associate-*l* 43.2%

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

      8. *-commutative 43.2%

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

      9. *-commutative 43.2%

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

      10. associate-*l* 42.8%

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

      11. unpow2 42.8%

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

      12. associate-*l* 43.9%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in l around -inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. mul-1-neg 41.7%

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

      3. *-commutative 41.7%

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

      4. *-commutative 41.7%

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

      5. unpow2 41.7%

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

      6. mul-1-neg 41.7%

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

      7. unsub-neg 41.7%

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

      8. associate-/l* 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in l around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. associate-*l/ 41.7%

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

      3. *-commutative 41.7%

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

      4. unpow2 41.7%

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

      5. *-commutative 41.7%

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

      6. distribute-neg-frac 41.7%

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

      7. distribute-rgt-neg-out 41.7%

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

      8. associate-/l* 42.6%

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

      9. neg-sub0 42.6%

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

      10. associate--r- 42.6%

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

      11. metadata-eval 42.6%

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

      12. *-commutative 42.6%

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

   9. Simplified 42.6%

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

   if -2.7000000000000001e-17 < l < -2.1e-88 or 4.20000000000000036e-40 < l < 1.7e45

   1. Initial program 55.3%

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

      1. associate-*l* 58.8%

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

      2. sub-neg 58.8%

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

      3. associate--l+ 58.8%

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

      4. *-commutative 58.8%

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

      5. distribute-rgt-neg-in 58.8%

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

      6. associate-*l/ 58.8%

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

      7. associate-*l* 58.8%

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

      8. *-commutative 58.8%

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

      9. *-commutative 58.8%

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

      10. associate-*l* 53.6%

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

      11. unpow2 53.6%

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

      12. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in l around -inf 59.4%

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

      1. mul-1-neg 59.4%

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

      2. unsub-neg 59.4%

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

      3. *-commutative 59.4%

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

      4. associate-/l* 59.4%

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

      5. mul-1-neg 59.4%

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

      6. unsub-neg 59.4%

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

      7. associate-/l* 62.4%

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

      8. *-commutative 62.4%

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

      9. unpow2 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in n around 0 58.0%

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

      1. associate-/l* 58.1%

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

      2. unpow2 58.1%

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

   9. Simplified 58.1%

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

   if -2.1e-88 < l < -1.09999999999999997e-106

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-*l* 68.4%

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

      3. associate-*l* 68.4%

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

      4. associate-*l* 68.4%

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

      5. sub-neg 68.4%

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

      6. +-commutative 68.4%

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

      7. *-commutative 68.4%

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

      8. distribute-rgt-neg-in 68.4%

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

      9. fma-def 68.4%

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

      10. associate-*r/ 68.4%

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

      11. metadata-eval 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in U* around inf 76.8%

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

      1. associate-*r/ 76.8%

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

      2. *-commutative 76.8%

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

      3. unpow2 76.8%

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

      4. *-commutative 76.8%

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

      5. unpow2 76.8%

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

   6. Simplified 76.8%

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

   if -1.09999999999999997e-106 < l < 2.05000000000000005e-135

   1. Initial program 68.4%

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

      1. associate-*l* 71.8%

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

      2. sub-neg 71.8%

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

      3. associate--l+ 71.8%

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

      4. *-commutative 71.8%

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

      5. distribute-rgt-neg-in 71.8%

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

      6. associate-*l/ 71.8%

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

      7. associate-*l* 71.8%

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

      8. *-commutative 71.8%

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

      9. *-commutative 71.8%

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

      10. associate-*l* 63.2%

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

      11. unpow2 63.2%

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

      12. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in t around inf 65.6%

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

      1. pow1/2 66.9%

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

      2. associate-*l* 66.9%

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

      3. *-commutative 66.9%

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

   6. Applied egg-rr 66.9%

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

   if 2.05000000000000005e-135 < l < 4.20000000000000036e-40

   1. Initial program 56.5%

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

      1. associate-*l* 45.3%

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

      2. sub-neg 45.3%

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

      3. associate--l+ 45.3%

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

      4. *-commutative 45.3%

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

      5. distribute-rgt-neg-in 45.3%

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

      6. associate-*l/ 45.3%

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

      7. associate-*l* 45.3%

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

      8. *-commutative 45.3%

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

      9. *-commutative 45.3%

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

      10. associate-*l* 40.9%

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

      11. unpow2 40.9%

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

      12. associate-*l* 41.3%

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

   3. Simplified 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. associate-*l* 41.3%

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

   5. Applied egg-rr 41.3%

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

      1. *-lft-identity 41.3%

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

      2. associate-*r* 52.6%

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

      3. *-commutative 52.6%

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

      4. associate-*r* 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in n around 0 48.6%

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

      1. unpow2 48.6%

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

   10. Simplified 48.6%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right)\right)}{Om \cdot Om}}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-135}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \end{array} \]

Alternative 7: 56.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-204} \lor \neg \left(n \leq 5.8 \cdot 10^{-195}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.15e-204) (not (<= n 5.8e-195)))
   (sqrt
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (* l -2.0) (/ (* n (* l U*)) Om)) (/ Om (* n (* U l)))))))
   (sqrt
    (*
     2.0
     (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ (* U (* n l)) Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.15e-204) || !(n <= 5.8e-195)) {
		tmp = sqrt((2.0 * ((n * (U * t)) + (((l * -2.0) + ((n * (l * U_42_)) / Om)) / (Om / (n * (U * l)))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * 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 ((n <= (-1.15d-204)) .or. (.not. (n <= 5.8d-195))) then
        tmp = sqrt((2.0d0 * ((n * (u * t)) + (((l * (-2.0d0)) + ((n * (l * u_42)) / om)) / (om / (n * (u * l)))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - ((u * (n * 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 ((n <= -1.15e-204) || !(n <= 5.8e-195)) {
		tmp = Math.sqrt((2.0 * ((n * (U * t)) + (((l * -2.0) + ((n * (l * U_42_)) / Om)) / (Om / (n * (U * l)))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.15e-204) or not (n <= 5.8e-195):
		tmp = math.sqrt((2.0 * ((n * (U * t)) + (((l * -2.0) + ((n * (l * U_42_)) / Om)) / (Om / (n * (U * l)))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.15e-204) || !(n <= 5.8e-195))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om)) / Float64(Om / Float64(n * Float64(U * l)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(Float64(U * Float64(n * l)) / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.15e-204) || ~((n <= 5.8e-195)))
		tmp = sqrt((2.0 * ((n * (U * t)) + (((l * -2.0) + ((n * (l * U_42_)) / Om)) / (Om / (n * (U * l)))))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.15e-204], N[Not[LessEqual[n, 5.8e-195]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.15e-204 or 5.8000000000000003e-195 < n

   1. Initial program 55.5%

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

      1. associate-*l* 56.8%

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

      2. sub-neg 56.8%

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

      3. associate--l+ 56.8%

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

      4. *-commutative 56.8%

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

      5. distribute-rgt-neg-in 56.8%

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

      6. associate-*l/ 58.8%

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

      7. associate-*l* 58.8%

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

      8. *-commutative 58.8%

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

      9. *-commutative 58.8%

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

      10. associate-*l* 53.7%

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

      11. unpow2 53.7%

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

      12. associate-*l* 54.3%

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

   3. Simplified 58.4%

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

      1. *-un-lft-identity 58.4%

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

      2. associate-*l* 58.4%

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

   5. Applied egg-rr 58.4%

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

      1. *-lft-identity 58.4%

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

      2. associate-*r* 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-*r* 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in t around 0 59.9%

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

      1. associate-/l* 61.9%

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

      2. +-commutative 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*r* 60.8%

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

      5. *-commutative 60.8%

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

   10. Simplified 60.8%

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

   11. Taylor expanded in U* around inf 62.4%

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

   if -1.15e-204 < n < 5.8000000000000003e-195

   1. Initial program 38.8%

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

      1. associate-*l* 32.3%

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

      2. sub-neg 32.3%

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

      3. associate--l+ 32.3%

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

      4. *-commutative 32.3%

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

      5. distribute-rgt-neg-in 32.3%

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

      6. associate-*l/ 41.9%

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

      7. associate-*l* 41.9%

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

      8. *-commutative 41.9%

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

      9. *-commutative 41.9%

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

      10. associate-*l* 41.8%

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

      11. unpow2 41.8%

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

      12. associate-*l* 45.8%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in U* around 0 34.0%

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

      1. associate-*r* 36.5%

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

      2. +-commutative 36.5%

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

   6. Simplified 56.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-204} \lor \neg \left(n \leq 5.8 \cdot 10^{-195}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 56.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -4.4 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -4.4e+99)
   (sqrt
    (* 2.0 (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ (* U (* n l)) Om)))))))))
   (sqrt
    (*
     2.0
     (+
      (/ (+ (* l -2.0) (/ (* (- U* U) (* n l)) Om)) (/ Om (* n (* U l))))
      (* n (* U t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -4.4e+99) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	} else {
		tmp = sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * 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 (u <= (-4.4d+99)) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - ((u * (n * l)) / om)))))))))
    else
        tmp = sqrt((2.0d0 * ((((l * (-2.0d0)) + (((u_42 - u) * (n * l)) / om)) / (om / (n * (u * l)))) + (n * (u * 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 (U <= -4.4e+99) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	} else {
		tmp = Math.sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -4.4e+99:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))))
	else:
		tmp = math.sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -4.4e+99)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(Float64(U * Float64(n * l)) / Om)))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(l * -2.0) + Float64(Float64(Float64(U_42_ - U) * Float64(n * l)) / Om)) / Float64(Om / Float64(n * Float64(U * l)))) + Float64(n * Float64(U * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -4.4e+99)
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	else
		tmp = sqrt((2.0 * ((((l * -2.0) + (((U_42_ - U) * (n * l)) / Om)) / (Om / (n * (U * l)))) + (n * (U * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -4.4e+99], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -4.39999999999999956e99

   1. Initial program 59.0%

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

      1. associate-*l* 37.8%

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

      2. sub-neg 37.8%

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

      3. associate--l+ 37.8%

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

      4. *-commutative 37.8%

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

      5. distribute-rgt-neg-in 37.8%

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

      6. associate-*l/ 37.8%

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

      7. associate-*l* 37.8%

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

      8. *-commutative 37.8%

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

      9. *-commutative 37.8%

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

      10. associate-*l* 27.1%

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

      11. unpow2 27.1%

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

      12. associate-*l* 27.3%

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

   3. Simplified 27.7%

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

   4. Taylor expanded in U* around 0 26.1%

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

      1. associate-*r* 41.2%

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

      2. +-commutative 41.2%

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

   6. Simplified 53.7%

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

   if -4.39999999999999956e99 < U

   1. Initial program 51.2%

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

      1. associate-*l* 54.0%

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

      2. sub-neg 54.0%

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

      3. associate--l+ 54.0%

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

      4. *-commutative 54.0%

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

      5. distribute-rgt-neg-in 54.0%

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

      6. associate-*l/ 57.9%

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

      7. associate-*l* 57.9%

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

      8. *-commutative 57.9%

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

      9. *-commutative 57.9%

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

      10. associate-*l* 54.8%

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

      11. unpow2 54.8%

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

      12. associate-*l* 56.2%

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

   3. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. associate-*l* 59.9%

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

   5. Applied egg-rr 59.9%

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

      1. *-lft-identity 59.9%

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

      2. associate-*r* 56.7%

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

      3. *-commutative 56.7%

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

      4. associate-*r* 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in t around 0 58.2%

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

      1. associate-/l* 59.0%

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

      2. +-commutative 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-*r* 60.3%

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

      5. *-commutative 60.3%

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

   10. Simplified 60.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4.4 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\ell \cdot -2 + \frac{\left(U* - U\right) \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 9: 46.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-135}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (/
           (* (* n 2.0) (* U (* l l)))
           (/ Om (- -2.0 (* (- U U*) (/ n Om))))))))
   (if (<= l -3.2e-17)
     t_1
     (if (<= l 5.3e-135)
       (pow (* 2.0 (* n (* U t))) 0.5)
       (if (<= l 2.25e-39)
         (sqrt (* 2.0 (* (* n U) (+ t (* -2.0 (/ (* l l) Om))))))
         (if (<= l 3.8e+44)
           (sqrt (* (* n 2.0) (- (* U t) (* 2.0 (/ (* l l) (/ Om U))))))
           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 * (l * l))) / (Om / (-2.0 - ((U - U_42_) * (n / Om))))));
	double tmp;
	if (l <= -3.2e-17) {
		tmp = t_1;
	} else if (l <= 5.3e-135) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 2.25e-39) {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else if (l <= 3.8e+44) {
		tmp = sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	} 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 * (l * l))) / (om / ((-2.0d0) - ((u - u_42) * (n / om))))))
    if (l <= (-3.2d-17)) then
        tmp = t_1
    else if (l <= 5.3d-135) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 2.25d-39) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l * l) / om))))))
    else if (l <= 3.8d+44) then
        tmp = sqrt(((n * 2.0d0) * ((u * t) - (2.0d0 * ((l * l) / (om / u))))))
    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 * (l * l))) / (Om / (-2.0 - ((U - U_42_) * (n / Om))))));
	double tmp;
	if (l <= -3.2e-17) {
		tmp = t_1;
	} else if (l <= 5.3e-135) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 2.25e-39) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else if (l <= 3.8e+44) {
		tmp = Math.sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((n * 2.0) * (U * (l * l))) / (Om / (-2.0 - ((U - U_42_) * (n / Om))))))
	tmp = 0
	if l <= -3.2e-17:
		tmp = t_1
	elif l <= 5.3e-135:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 2.25e-39:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	elif l <= 3.8e+44:
		tmp = math.sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(n * 2.0) * Float64(U * Float64(l * l))) / Float64(Om / Float64(-2.0 - Float64(Float64(U - U_42_) * Float64(n / Om))))))
	tmp = 0.0
	if (l <= -3.2e-17)
		tmp = t_1;
	elseif (l <= 5.3e-135)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 2.25e-39)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	elseif (l <= 3.8e+44)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) - Float64(2.0 * Float64(Float64(l * l) / Float64(Om / U))))));
	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 * (l * l))) / (Om / (-2.0 - ((U - U_42_) * (n / Om))))));
	tmp = 0.0;
	if (l <= -3.2e-17)
		tmp = t_1;
	elseif (l <= 5.3e-135)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 2.25e-39)
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	elseif (l <= 3.8e+44)
		tmp = sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	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[(n * 2.0), $MachinePrecision] * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(-2.0 - N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.2e-17], t$95$1, If[LessEqual[l, 5.3e-135], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.25e-39], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e+44], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] - N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.2000000000000002e-17 or 3.8000000000000002e44 < l

   1. Initial program 36.6%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. associate--l+ 34.9%

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

      4. *-commutative 34.9%

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

      5. distribute-rgt-neg-in 34.9%

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

      6. associate-*l/ 43.2%

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

      7. associate-*l* 43.2%

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

      8. *-commutative 43.2%

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

      9. *-commutative 43.2%

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

      10. associate-*l* 42.8%

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

      11. unpow2 42.8%

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

      12. associate-*l* 43.9%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in l around -inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. mul-1-neg 41.7%

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

      3. *-commutative 41.7%

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

      4. *-commutative 41.7%

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

      5. unpow2 41.7%

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

      6. mul-1-neg 41.7%

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

      7. unsub-neg 41.7%

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

      8. associate-/l* 41.7%

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

   6. Simplified 41.7%

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

      1. *-un-lft-identity 41.7%

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

      2. associate-*l* 41.7%

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

      3. distribute-rgt-neg-in 41.7%

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

      4. associate-/r/ 41.7%

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

   8. Applied egg-rr 41.7%

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

      1. *-lft-identity 41.7%

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

      2. associate-*r* 41.7%

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

      3. associate-/l* 42.6%

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

      4. unpow2 42.6%

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

      5. *-commutative 42.6%

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

      6. associate-*r/ 42.7%

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

   10. Simplified 42.7%

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

   if -3.2000000000000002e-17 < l < 5.3e-135

   1. Initial program 66.2%

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

      1. associate-*l* 69.8%

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

      2. sub-neg 69.8%

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

      3. associate--l+ 69.8%

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

      4. *-commutative 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. associate-*l/ 69.8%

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

      7. associate-*l* 69.8%

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

      8. *-commutative 69.8%

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

      9. *-commutative 69.8%

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

      10. associate-*l* 61.6%

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

      11. unpow2 61.6%

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

      12. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in t around inf 61.6%

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

      1. pow1/2 62.6%

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

      2. associate-*l* 62.6%

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

      3. *-commutative 62.6%

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

   6. Applied egg-rr 62.6%

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

   if 5.3e-135 < l < 2.25e-39

   1. Initial program 56.5%

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

      1. associate-*l* 45.3%

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

      2. sub-neg 45.3%

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

      3. associate--l+ 45.3%

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

      4. *-commutative 45.3%

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

      5. distribute-rgt-neg-in 45.3%

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

      6. associate-*l/ 45.3%

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

      7. associate-*l* 45.3%

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

      8. *-commutative 45.3%

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

      9. *-commutative 45.3%

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

      10. associate-*l* 40.9%

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

      11. unpow2 40.9%

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

      12. associate-*l* 41.3%

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

   3. Simplified 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. associate-*l* 41.3%

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

   5. Applied egg-rr 41.3%

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

      1. *-lft-identity 41.3%

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

      2. associate-*r* 52.6%

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

      3. *-commutative 52.6%

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

      4. associate-*r* 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in n around 0 48.6%

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

      1. unpow2 48.6%

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

   10. Simplified 48.6%

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

   if 2.25e-39 < l < 3.8000000000000002e44

   1. Initial program 55.3%

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

      1. associate-*l* 55.9%

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

      2. sub-neg 55.9%

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

      3. associate--l+ 55.9%

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

      4. *-commutative 55.9%

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

      5. distribute-rgt-neg-in 55.9%

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

      6. associate-*l/ 55.9%

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

      7. associate-*l* 55.9%

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

      8. *-commutative 55.9%

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

      9. *-commutative 55.9%

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

      10. associate-*l* 55.7%

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

      11. unpow2 55.7%

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

      12. associate-*l* 62.4%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in l around -inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. unsub-neg 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-/l* 61.8%

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

      5. mul-1-neg 61.8%

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

      6. unsub-neg 61.8%

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

      7. associate-/l* 61.8%

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

      8. *-commutative 61.8%

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

      9. unpow2 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in n around 0 47.7%

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

      1. associate-/l* 47.9%

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

      2. unpow2 47.9%

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

   9. Simplified 47.9%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-135}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{\frac{Om}{-2 - \left(U - U*\right) \cdot \frac{n}{Om}}}}\\ \end{array} \]

Alternative 10: 49.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n}{\frac{Om}{U* - U}} - 2\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-136}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* n 2.0)
           (/ (* (* l (* U l)) (- (/ n (/ Om (- U* U))) 2.0)) Om)))))
   (if (<= l -1.55e-17)
     t_1
     (if (<= l 1.1e-136)
       (pow (* 2.0 (* n (* U t))) 0.5)
       (if (<= l 7.3e-40)
         (sqrt (* 2.0 (* (* n U) (+ t (* -2.0 (/ (* l l) Om))))))
         (if (<= l 8e+44)
           (sqrt (* (* n 2.0) (- (* U t) (* 2.0 (/ (* l l) (/ Om U))))))
           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) * (((l * (U * l)) * ((n / (Om / (U_42_ - U))) - 2.0)) / Om)));
	double tmp;
	if (l <= -1.55e-17) {
		tmp = t_1;
	} else if (l <= 1.1e-136) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 7.3e-40) {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else if (l <= 8e+44) {
		tmp = sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	} 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) * (((l * (u * l)) * ((n / (om / (u_42 - u))) - 2.0d0)) / om)))
    if (l <= (-1.55d-17)) then
        tmp = t_1
    else if (l <= 1.1d-136) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 7.3d-40) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l * l) / om))))))
    else if (l <= 8d+44) then
        tmp = sqrt(((n * 2.0d0) * ((u * t) - (2.0d0 * ((l * l) / (om / u))))))
    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) * (((l * (U * l)) * ((n / (Om / (U_42_ - U))) - 2.0)) / Om)));
	double tmp;
	if (l <= -1.55e-17) {
		tmp = t_1;
	} else if (l <= 1.1e-136) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 7.3e-40) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else if (l <= 8e+44) {
		tmp = Math.sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((n * 2.0) * (((l * (U * l)) * ((n / (Om / (U_42_ - U))) - 2.0)) / Om)))
	tmp = 0
	if l <= -1.55e-17:
		tmp = t_1
	elif l <= 1.1e-136:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 7.3e-40:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	elif l <= 8e+44:
		tmp = math.sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))))
	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(Float64(Float64(l * Float64(U * l)) * Float64(Float64(n / Float64(Om / Float64(U_42_ - U))) - 2.0)) / Om)))
	tmp = 0.0
	if (l <= -1.55e-17)
		tmp = t_1;
	elseif (l <= 1.1e-136)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 7.3e-40)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	elseif (l <= 8e+44)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) - Float64(2.0 * Float64(Float64(l * l) / Float64(Om / U))))));
	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) * (((l * (U * l)) * ((n / (Om / (U_42_ - U))) - 2.0)) / Om)));
	tmp = 0.0;
	if (l <= -1.55e-17)
		tmp = t_1;
	elseif (l <= 1.1e-136)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 7.3e-40)
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	elseif (l <= 8e+44)
		tmp = sqrt(((n * 2.0) * ((U * t) - (2.0 * ((l * l) / (Om / U))))));
	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[(N[(N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.55e-17], t$95$1, If[LessEqual[l, 1.1e-136], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 7.3e-40], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8e+44], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] - N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(Om / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.5499999999999999e-17 or 8.0000000000000007e44 < l

   1. Initial program 36.6%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. associate--l+ 34.9%

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

      4. *-commutative 34.9%

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

      5. distribute-rgt-neg-in 34.9%

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

      6. associate-*l/ 43.2%

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

      7. associate-*l* 43.2%

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

      8. *-commutative 43.2%

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

      9. *-commutative 43.2%

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

      10. associate-*l* 42.8%

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

      11. unpow2 42.8%

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

      12. associate-*l* 43.9%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in l around -inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. mul-1-neg 41.7%

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

      3. *-commutative 41.7%

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

      4. *-commutative 41.7%

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

      5. unpow2 41.7%

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

      6. mul-1-neg 41.7%

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

      7. unsub-neg 41.7%

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

      8. associate-/l* 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in U around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. unpow2 41.7%

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

      3. associate-*l* 44.5%

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

      4. *-commutative 44.5%

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

      5. *-commutative 44.5%

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

   9. Simplified 44.5%

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

   if -1.5499999999999999e-17 < l < 1.1000000000000001e-136

   1. Initial program 66.2%

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

      1. associate-*l* 69.8%

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

      2. sub-neg 69.8%

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

      3. associate--l+ 69.8%

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

      4. *-commutative 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. associate-*l/ 69.8%

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

      7. associate-*l* 69.8%

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

      8. *-commutative 69.8%

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

      9. *-commutative 69.8%

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

      10. associate-*l* 61.6%

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

      11. unpow2 61.6%

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

      12. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in t around inf 61.6%

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

      1. pow1/2 62.6%

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

      2. associate-*l* 62.6%

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

      3. *-commutative 62.6%

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

   6. Applied egg-rr 62.6%

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

   if 1.1000000000000001e-136 < l < 7.30000000000000005e-40

   1. Initial program 56.5%

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

      1. associate-*l* 45.3%

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

      2. sub-neg 45.3%

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

      3. associate--l+ 45.3%

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

      4. *-commutative 45.3%

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

      5. distribute-rgt-neg-in 45.3%

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

      6. associate-*l/ 45.3%

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

      7. associate-*l* 45.3%

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

      8. *-commutative 45.3%

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

      9. *-commutative 45.3%

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

      10. associate-*l* 40.9%

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

      11. unpow2 40.9%

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

      12. associate-*l* 41.3%

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

   3. Simplified 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. associate-*l* 41.3%

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

   5. Applied egg-rr 41.3%

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

      1. *-lft-identity 41.3%

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

      2. associate-*r* 52.6%

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

      3. *-commutative 52.6%

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

      4. associate-*r* 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in n around 0 48.6%

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

      1. unpow2 48.6%

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

   10. Simplified 48.6%

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

   if 7.30000000000000005e-40 < l < 8.0000000000000007e44

   1. Initial program 55.3%

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

      1. associate-*l* 55.9%

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

      2. sub-neg 55.9%

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

      3. associate--l+ 55.9%

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

      4. *-commutative 55.9%

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

      5. distribute-rgt-neg-in 55.9%

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

      6. associate-*l/ 55.9%

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

      7. associate-*l* 55.9%

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

      8. *-commutative 55.9%

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

      9. *-commutative 55.9%

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

      10. associate-*l* 55.7%

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

      11. unpow2 55.7%

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

      12. associate-*l* 62.4%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in l around -inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. unsub-neg 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-/l* 61.8%

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

      5. mul-1-neg 61.8%

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

      6. unsub-neg 61.8%

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

      7. associate-/l* 61.8%

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

      8. *-commutative 61.8%

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

      9. unpow2 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in n around 0 47.7%

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

      1. associate-/l* 47.9%

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

      2. unpow2 47.9%

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

   9. Simplified 47.9%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n}{\frac{Om}{U* - U}} - 2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-136}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t - 2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n}{\frac{Om}{U* - U}} - 2\right)}{Om}}\\ \end{array} \]

Alternative 11: 48.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-37} \lor \neg \left(n \leq 5.2 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.75e-37) (not (<= n 5.2e-58)))
   (sqrt (* (* n 2.0) (+ (* U t) (/ (* (* n (* l l)) (* U U*)) (* Om Om)))))
   (sqrt
    (*
     2.0
     (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ n (/ Om (* U l)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.75e-37) || !(n <= 5.2e-58)) {
		tmp = sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	}
	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 <= (-1.75d-37)) .or. (.not. (n <= 5.2d-58))) then
        tmp = sqrt(((n * 2.0d0) * ((u * t) + (((n * (l * l)) * (u * u_42)) / (om * om)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - (n / (om / (u * l)))))))))))
    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 <= -1.75e-37) || !(n <= 5.2e-58)) {
		tmp = Math.sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.75e-37) or not (n <= 5.2e-58):
		tmp = math.sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.75e-37) || !(n <= 5.2e-58))
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) + Float64(Float64(Float64(n * Float64(l * l)) * Float64(U * U_42_)) / Float64(Om * Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(n / Float64(Om / Float64(U * l)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.75e-37) || ~((n <= 5.2e-58)))
		tmp = sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.75e-37], N[Not[LessEqual[n, 5.2e-58]], $MachinePrecision]], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(n / N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.7500000000000001e-37 or 5.20000000000000013e-58 < n

   1. Initial program 56.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 57.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 50.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around -inf 50.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + -1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

      2. unsub-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U - \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

      3. *-commutative 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{U \cdot t} - \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]

      4. associate-/l* 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

      5. mul-1-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)} \]

      6. unsub-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{\color{blue}{2 - \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)} \]

      7. associate-/l* 53.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)} \]

      8. *-commutative 53.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 - \frac{n}{\frac{Om}{U* - U}}}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}\right)} \]

      9. unpow2 53.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 - \frac{n}{\frac{Om}{U* - U}}}{\frac{Om}{U \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}\right)} \]

   6. Simplified 53.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t - \frac{2 - \frac{n}{\frac{Om}{U* - U}}}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}} \]

   7. Taylor expanded in U* around inf 50.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 50.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{\frac{-1 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)\right)}{{Om}^{2}}}\right)} \]

      2. mul-1-neg 50.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{\color{blue}{-n \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}}{{Om}^{2}}\right)} \]

      3. associate-*r* 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot U*\right)}}{{Om}^{2}}\right)} \]

      4. unpow2 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot U*\right)}{{Om}^{2}}\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(U* \cdot U\right)}}{{Om}^{2}}\right)} \]

      6. unpow2 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   9. Simplified 49.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{\frac{-\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om}}\right)} \]

   if -1.7500000000000001e-37 < n < 5.20000000000000013e-58

   1. Initial program 45.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.6%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 52.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 52.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 55.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 56.0%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      2. associate-*l* 56.0%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

   5. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-lft-identity 56.0%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

      2. associate-*r* 55.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      3. *-commutative 55.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)\right)} \]

      4. associate-*r* 55.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}\right) \cdot \frac{\ell}{Om}\right)\right)} \]

   7. Simplified 55.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}} \]

   8. Taylor expanded in U* around 0 42.1%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(\left(t + \frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om}\right) \cdot U\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 44.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t + \frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om}\right)\right) \cdot U\right)}} \]

      2. associate-/l* 53.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + \color{blue}{\frac{\ell}{\frac{Om}{-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell}}}\right)\right) \cdot U\right)} \]

      3. +-commutative 53.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\color{blue}{-2 \cdot \ell + -1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om}}}}\right)\right) \cdot U\right)} \]

      4. mul-1-neg 53.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{-2 \cdot \ell + \color{blue}{\left(-\frac{n \cdot \left(\ell \cdot U\right)}{Om}\right)}}}\right)\right) \cdot U\right)} \]

      5. unsub-neg 53.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\color{blue}{-2 \cdot \ell - \frac{n \cdot \left(\ell \cdot U\right)}{Om}}}}\right)\right) \cdot U\right)} \]

      6. *-commutative 53.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\color{blue}{\ell \cdot -2} - \frac{n \cdot \left(\ell \cdot U\right)}{Om}}}\right)\right) \cdot U\right)} \]

      7. associate-/l* 53.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \color{blue}{\frac{n}{\frac{Om}{\ell \cdot U}}}}}\right)\right) \cdot U\right)} \]

   10. Simplified 53.3%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{\ell \cdot U}}}}\right)\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-37} \lor \neg \left(n \leq 5.2 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 49.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.78 \cdot 10^{-37} \lor \neg \left(n \leq 2.85 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.78e-37) (not (<= n 2.85e-58)))
   (sqrt (* (* n 2.0) (+ (* U t) (/ (* (* n (* l l)) (* U U*)) (* Om Om)))))
   (sqrt
    (*
     2.0
     (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ (* U (* n l)) Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.78e-37) || !(n <= 2.85e-58)) {
		tmp = sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * 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 ((n <= (-1.78d-37)) .or. (.not. (n <= 2.85d-58))) then
        tmp = sqrt(((n * 2.0d0) * ((u * t) + (((n * (l * l)) * (u * u_42)) / (om * om)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - ((u * (n * 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 ((n <= -1.78e-37) || !(n <= 2.85e-58)) {
		tmp = Math.sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.78e-37) or not (n <= 2.85e-58):
		tmp = math.sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.78e-37) || !(n <= 2.85e-58))
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) + Float64(Float64(Float64(n * Float64(l * l)) * Float64(U * U_42_)) / Float64(Om * Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(Float64(U * Float64(n * l)) / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.78e-37) || ~((n <= 2.85e-58)))
		tmp = sqrt(((n * 2.0) * ((U * t) + (((n * (l * l)) * (U * U_42_)) / (Om * Om)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.78e-37], N[Not[LessEqual[n, 2.85e-58]], $MachinePrecision]], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.78000000000000005e-37 or 2.85000000000000016e-58 < n

   1. Initial program 56.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 57.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 57.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 50.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around -inf 50.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U + -1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U + \color{blue}{\left(-\frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}\right)} \]

      2. unsub-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U - \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]

      3. *-commutative 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\color{blue}{U \cdot t} - \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)} \]

      4. associate-/l* 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{\frac{2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

      5. mul-1-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)} \]

      6. unsub-neg 50.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{\color{blue}{2 - \frac{n \cdot \left(U* - U\right)}{Om}}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)} \]

      7. associate-/l* 53.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}}{\frac{Om}{{\ell}^{2} \cdot U}}\right)} \]

      8. *-commutative 53.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 - \frac{n}{\frac{Om}{U* - U}}}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}\right)} \]

      9. unpow2 53.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{2 - \frac{n}{\frac{Om}{U* - U}}}{\frac{Om}{U \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}\right)} \]

   6. Simplified 53.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t - \frac{2 - \frac{n}{\frac{Om}{U* - U}}}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}} \]

   7. Taylor expanded in U* around inf 50.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 50.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{\frac{-1 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)\right)}{{Om}^{2}}}\right)} \]

      2. mul-1-neg 50.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{\color{blue}{-n \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}}{{Om}^{2}}\right)} \]

      3. associate-*r* 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot U*\right)}}{{Om}^{2}}\right)} \]

      4. unpow2 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot U*\right)}{{Om}^{2}}\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(U* \cdot U\right)}}{{Om}^{2}}\right)} \]

      6. unpow2 49.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{-\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   9. Simplified 49.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \color{blue}{\frac{-\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om}}\right)} \]

   if -1.78000000000000005e-37 < n < 2.85000000000000016e-58

   1. Initial program 45.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.6%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 52.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 52.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 55.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U* around 0 42.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(\left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right) \cdot U\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 44.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right)\right) \cdot U\right)}} \]

      2. +-commutative 44.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(t + \frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om}\right)}\right) \cdot U\right)} \]

   6. Simplified 58.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{\left(n \cdot \ell\right) \cdot U}{Om}}}\right)\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.78 \cdot 10^{-37} \lor \neg \left(n \leq 2.85 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \]

Alternative 13: 56.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-230} \lor \neg \left(n \leq 2 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.9e-230) (not (<= n 2e-58)))
   (sqrt
    (*
     (* n 2.0)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
   (sqrt
    (*
     2.0
     (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ (* U (* n l)) Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.9e-230) || !(n <= 2e-58)) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * 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 ((n <= (-1.9d-230)) .or. (.not. (n <= 2d-58))) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - ((u * (n * 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 ((n <= -1.9e-230) || !(n <= 2e-58)) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.9e-230) or not (n <= 2e-58):
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.9e-230) || !(n <= 2e-58))
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(Float64(U * Float64(n * l)) / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.9e-230) || ~((n <= 2e-58)))
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - ((U * (n * l)) / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.9e-230], N[Not[LessEqual[n, 2e-58]], $MachinePrecision]], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.8999999999999999e-230 or 2.0000000000000001e-58 < n

   1. Initial program 55.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 58.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 58.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 58.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 58.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 59.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 59.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 59.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 59.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 53.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 53.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 53.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U around 0 60.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

   if -1.8999999999999999e-230 < n < 2.0000000000000001e-58

   1. Initial program 43.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 36.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 36.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 36.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 36.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 36.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 46.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 46.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 46.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 46.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 45.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 45.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U* around 0 37.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(\left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right) \cdot U\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 41.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(\frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om} + t\right)\right) \cdot U\right)}} \]

      2. +-commutative 41.8%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(t + \frac{\ell \cdot \left(-1 \cdot \frac{n \cdot \left(\ell \cdot U\right)}{Om} + -2 \cdot \ell\right)}{Om}\right)}\right) \cdot U\right)} \]

   6. Simplified 58.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{\left(n \cdot \ell\right) \cdot U}{Om}}}\right)\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-230} \lor \neg \left(n \leq 2 \cdot 10^{-58}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{U \cdot \left(n \cdot \ell\right)}{Om}}}\right)\right)\right)}\\ \end{array} \]

Alternative 14: 43.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+161}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-12} \lor \neg \left(t \leq -1.8 \cdot 10^{-52}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\ell \cdot \left(-\ell\right)}{\frac{Om}{U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5e+161)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (or (<= t -8.5e-12) (not (<= t -1.8e-52)))
     (sqrt (* 2.0 (* (* n U) (+ t (* -2.0 (/ (* l l) Om))))))
     (sqrt
      (* (* n 2.0) (/ (* l (- l)) (/ Om (* U (- 2.0 (/ n (/ Om U*)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e+161) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if ((t <= -8.5e-12) || !(t <= -1.8e-52)) {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt(((n * 2.0) * ((l * -l) / (Om / (U * (2.0 - (n / (Om / U_42_))))))));
	}
	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 <= (-5d+161)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if ((t <= (-8.5d-12)) .or. (.not. (t <= (-1.8d-52)))) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l * l) / om))))))
    else
        tmp = sqrt(((n * 2.0d0) * ((l * -l) / (om / (u * (2.0d0 - (n / (om / u_42))))))))
    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 <= -5e+161) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if ((t <= -8.5e-12) || !(t <= -1.8e-52)) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * ((l * -l) / (Om / (U * (2.0 - (n / (Om / U_42_))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5e+161:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif (t <= -8.5e-12) or not (t <= -1.8e-52):
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt(((n * 2.0) * ((l * -l) / (Om / (U * (2.0 - (n / (Om / U_42_))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5e+161)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif ((t <= -8.5e-12) || !(t <= -1.8e-52))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(l * Float64(-l)) / Float64(Om / Float64(U * Float64(2.0 - Float64(n / Float64(Om / U_42_))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5e+161)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif ((t <= -8.5e-12) || ~((t <= -1.8e-52)))
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt(((n * 2.0) * ((l * -l) / (Om / (U * (2.0 - (n / (Om / U_42_))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5e+161], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[Or[LessEqual[t, -8.5e-12], N[Not[LessEqual[t, -1.8e-52]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l * (-l)), $MachinePrecision] / N[(Om / N[(U * N[(2.0 - N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.9999999999999997e161

   1. Initial program 44.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 60.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 61.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 64.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

      2. associate-*l* 64.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

      3. *-commutative 64.3%

         \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]

   if -4.9999999999999997e161 < t < -8.4999999999999997e-12 or -1.79999999999999994e-52 < t

   1. Initial program 53.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.5%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 51.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 50.0%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 50.0%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 51.0%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 53.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 53.5%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      2. associate-*l* 53.5%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

   5. Applied egg-rr 53.5%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-lft-identity 53.5%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

      2. associate-*r* 56.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      3. *-commutative 56.7%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)\right)} \]

      4. associate-*r* 60.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}\right) \cdot \frac{\ell}{Om}\right)\right)} \]

   7. Simplified 60.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}} \]

   8. Taylor expanded in n around 0 45.1%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 45.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) \cdot \left(n \cdot U\right)\right)} \]

   10. Simplified 45.1%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]

   if -8.4999999999999997e-12 < t < -1.79999999999999994e-52

   1. Initial program 51.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 61.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 61.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 60.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 60.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 61.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around -inf 61.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-1 \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)}{Om}}} \]

      2. mul-1-neg 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\color{blue}{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}}{Om}} \]

      3. *-commutative 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left({\ell}^{2} \cdot U\right) \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{Om}} \]

      4. *-commutative 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(U \cdot {\ell}^{2}\right)} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}} \]

      5. unpow2 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}} \]

      6. mul-1-neg 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{Om}} \]

      7. unsub-neg 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{Om}} \]

      8. associate-/l* 61.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{Om}} \]

   6. Simplified 61.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{Om}}} \]

   7. Taylor expanded in U around 0 80.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{{\ell}^{2} \cdot \left(\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U\right)}{Om}\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg 80.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{{\ell}^{2} \cdot \left(\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U\right)}{Om}\right)}} \]

      2. associate-/l* 80.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\color{blue}{\frac{{\ell}^{2}}{\frac{Om}{\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U}}}\right)} \]

      3. unpow2 80.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\frac{\color{blue}{\ell \cdot \ell}}{\frac{Om}{\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U}}\right)} \]

      4. *-commutative 80.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\frac{\ell \cdot \ell}{\frac{Om}{\color{blue}{U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)}}}\right)} \]

      5. associate-/l* 80.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(-\frac{\ell \cdot \ell}{\frac{Om}{U \cdot \left(2 - \color{blue}{\frac{n}{\frac{Om}{U*}}}\right)}}\right)} \]

   9. Simplified 80.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-\frac{\ell \cdot \ell}{\frac{Om}{U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+161}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-12} \lor \neg \left(t \leq -1.8 \cdot 10^{-52}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\ell \cdot \left(-\ell\right)}{\frac{Om}{U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)}}}\\ \end{array} \]

Alternative 15: 43.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-12} \lor \neg \left(t \leq -5.2 \cdot 10^{-47}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{n}{\frac{Om}{U} \cdot \frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -1.3e+162)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (or (<= t -2.75e-12) (not (<= t -5.2e-47)))
     (sqrt (* 2.0 (* (* n U) (+ t (* -2.0 (/ (* l l) Om))))))
     (sqrt (* (* n 2.0) (/ n (* (/ Om U) (/ Om (* U* (* l l))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.3e+162) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if ((t <= -2.75e-12) || !(t <= -5.2e-47)) {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt(((n * 2.0) * (n / ((Om / U) * (Om / (U_42_ * (l * l)))))));
	}
	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 <= (-1.3d+162)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if ((t <= (-2.75d-12)) .or. (.not. (t <= (-5.2d-47)))) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l * l) / om))))))
    else
        tmp = sqrt(((n * 2.0d0) * (n / ((om / u) * (om / (u_42 * (l * l)))))))
    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 <= -1.3e+162) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if ((t <= -2.75e-12) || !(t <= -5.2e-47)) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (n / ((Om / U) * (Om / (U_42_ * (l * l)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -1.3e+162:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif (t <= -2.75e-12) or not (t <= -5.2e-47):
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt(((n * 2.0) * (n / ((Om / U) * (Om / (U_42_ * (l * l)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -1.3e+162)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif ((t <= -2.75e-12) || !(t <= -5.2e-47))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(n / Float64(Float64(Om / U) * Float64(Om / Float64(U_42_ * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.3e+162)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif ((t <= -2.75e-12) || ~((t <= -5.2e-47)))
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt(((n * 2.0) * (n / ((Om / U) * (Om / (U_42_ * (l * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -1.3e+162], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[Or[LessEqual[t, -2.75e-12], N[Not[LessEqual[t, -5.2e-47]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(n / N[(N[(Om / U), $MachinePrecision] * N[(Om / N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3e162

   1. Initial program 44.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 60.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 61.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 64.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

      2. associate-*l* 64.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

      3. *-commutative 64.3%

         \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]

   if -1.3e162 < t < -2.7500000000000002e-12 or -5.2e-47 < t

   1. Initial program 53.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 54.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 54.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 54.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 54.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 50.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 51.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 53.7%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      2. associate-*l* 53.7%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

   5. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-lft-identity 53.7%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

      2. associate-*r* 56.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      3. *-commutative 56.9%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)\right)} \]

      4. associate-*r* 60.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}\right) \cdot \frac{\ell}{Om}\right)\right)} \]

   7. Simplified 60.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}} \]

   8. Taylor expanded in n around 0 44.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 44.9%

         \[\leadsto \sqrt{2 \cdot \left(\left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) \cdot \left(n \cdot U\right)\right)} \]

   10. Simplified 44.9%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]

   if -2.7500000000000002e-12 < t < -5.2e-47

   1. Initial program 46.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.9%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 56.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 56.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 56.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around -inf 56.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-1 \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)\right)}{Om}}} \]

      2. mul-1-neg 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\color{blue}{-\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot \left({\ell}^{2} \cdot U\right)}}{Om}} \]

      3. *-commutative 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left({\ell}^{2} \cdot U\right) \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{Om}} \]

      4. *-commutative 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\color{blue}{\left(U \cdot {\ell}^{2}\right)} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}} \]

      5. unpow2 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}} \]

      6. mul-1-neg 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)}{Om}} \]

      7. unsub-neg 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}}{Om}} \]

      8. associate-/l* 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)}{Om}} \]

   6. Simplified 56.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{-\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}{Om}}} \]

   7. Taylor expanded in U* around inf 57.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{n \cdot \left({\ell}^{2} \cdot \left(U* \cdot U\right)\right)}{{Om}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot \left(U* \cdot U\right)}}}} \]

      2. unpow2 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\color{blue}{Om \cdot Om}}{{\ell}^{2} \cdot \left(U* \cdot U\right)}}} \]

      3. associate-*r* 56.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{Om \cdot Om}{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot U}}}} \]

      4. times-frac 67.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\color{blue}{\frac{Om}{{\ell}^{2} \cdot U*} \cdot \frac{Om}{U}}}} \]

      5. unpow2 67.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot U*} \cdot \frac{Om}{U}}} \]

   9. Simplified 67.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot U*} \cdot \frac{Om}{U}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-12} \lor \neg \left(t \leq -5.2 \cdot 10^{-47}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{n}{\frac{Om}{U} \cdot \frac{Om}{U* \cdot \left(\ell \cdot \ell\right)}}}\\ \end{array} \]

Alternative 16: 44.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8800 \lor \neg \left(\ell \leq 2.3 \cdot 10^{+45}\right):\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -8800.0) (not (<= l 2.3e+45)))
   (pow (* 2.0 (* n (* -2.0 (* U (/ (* l l) Om))))) 0.5)
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -8800.0) || !(l <= 2.3e+45)) {
		tmp = pow((2.0 * (n * (-2.0 * (U * ((l * l) / Om))))), 0.5);
	} else {
		tmp = pow((2.0 * (n * (U * 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 ((l <= (-8800.0d0)) .or. (.not. (l <= 2.3d+45))) then
        tmp = (2.0d0 * (n * ((-2.0d0) * (u * ((l * l) / om))))) ** 0.5d0
    else
        tmp = (2.0d0 * (n * (u * 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 ((l <= -8800.0) || !(l <= 2.3e+45)) {
		tmp = Math.pow((2.0 * (n * (-2.0 * (U * ((l * l) / Om))))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -8800.0) or not (l <= 2.3e+45):
		tmp = math.pow((2.0 * (n * (-2.0 * (U * ((l * l) / Om))))), 0.5)
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -8800.0) || !(l <= 2.3e+45))
		tmp = Float64(2.0 * Float64(n * Float64(-2.0 * Float64(U * Float64(Float64(l * l) / Om))))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -8800.0) || ~((l <= 2.3e+45)))
		tmp = (2.0 * (n * (-2.0 * (U * ((l * l) / Om))))) ^ 0.5;
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -8800.0], N[Not[LessEqual[l, 2.3e+45]], $MachinePrecision]], N[Power[N[(2.0 * N[(n * N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8800 or 2.30000000000000012e45 < l

   1. Initial program 35.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 42.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 42.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 42.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 42.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 42.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 42.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 43.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around inf 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 34.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

      2. unpow2 34.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      3. associate-/l* 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      4. unpow2 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U* - U}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      5. associate-*r/ 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)} \]

      6. metadata-eval 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{\color{blue}{2}}{Om}\right)\right)\right)} \]

   6. Simplified 34.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)\right)}} \]

   7. Taylor expanded in U* around 0 10.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left({\ell}^{2} \cdot \left(\left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow2 10.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)} \]

      2. *-commutative 10.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]

      3. mul-1-neg 10.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\frac{n \cdot U}{{Om}^{2}}\right)} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      4. associate-/l* 10.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\color{blue}{\frac{n}{\frac{{Om}^{2}}{U}}}\right) - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      5. unpow2 10.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U}}\right) - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      6. associate-*r/ 10.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{Om \cdot Om}{U}}\right) - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      7. metadata-eval 10.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{Om \cdot Om}{U}}\right) - \frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

   9. Simplified 10.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{Om \cdot Om}{U}}\right) - \frac{2}{Om}\right)\right)\right)\right)}} \]

   10. Taylor expanded in U around 0 24.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}\right)} \]
   11. Step-by-step derivation

       1. associate-/l* 21.8%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{Om}{U}}}\right)\right)} \]

       2. unpow2 21.8%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{Om}{U}}\right)\right)} \]

   12. Simplified 21.8%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\right)} \]
   13. Step-by-step derivation

       1. pow1/2 29.9%

          \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)\right)\right)}^{0.5}} \]

       2. associate-/r/ 32.8%

          \[\leadsto {\left(2 \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{Om} \cdot U\right)}\right)\right)\right)}^{0.5} \]

   14. Applied egg-rr 32.8%

       \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(-2 \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot U\right)\right)\right)\right)}^{0.5}} \]

   if -8800 < l < 2.30000000000000012e45

   1. Initial program 63.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 63.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 57.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 57.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 58.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 58.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 51.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 52.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

      2. associate-*l* 52.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

      3. *-commutative 52.3%

         \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8800 \lor \neg \left(\ell \leq 2.3 \cdot 10^{+45}\right):\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 17: 39.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5800 \lor \neg \left(\ell \leq 7.2 \cdot 10^{+50}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -5800.0) (not (<= l 7.2e+50)))
   (sqrt (* 2.0 (* -2.0 (/ n (/ Om (* U (* l l)))))))
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -5800.0) || !(l <= 7.2e+50)) {
		tmp = sqrt((2.0 * (-2.0 * (n / (Om / (U * (l * l)))))));
	} else {
		tmp = pow((2.0 * (n * (U * 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 ((l <= (-5800.0d0)) .or. (.not. (l <= 7.2d+50))) then
        tmp = sqrt((2.0d0 * ((-2.0d0) * (n / (om / (u * (l * l)))))))
    else
        tmp = (2.0d0 * (n * (u * 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 ((l <= -5800.0) || !(l <= 7.2e+50)) {
		tmp = Math.sqrt((2.0 * (-2.0 * (n / (Om / (U * (l * l)))))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -5800.0) or not (l <= 7.2e+50):
		tmp = math.sqrt((2.0 * (-2.0 * (n / (Om / (U * (l * l)))))))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -5800.0) || !(l <= 7.2e+50))
		tmp = sqrt(Float64(2.0 * Float64(-2.0 * Float64(n / Float64(Om / Float64(U * Float64(l * l)))))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -5800.0) || ~((l <= 7.2e+50)))
		tmp = sqrt((2.0 * (-2.0 * (n / (Om / (U * (l * l)))))));
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -5800.0], N[Not[LessEqual[l, 7.2e+50]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(-2.0 * N[(n / N[(Om / N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5800 or 7.19999999999999972e50 < l

   1. Initial program 35.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 42.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 42.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 43.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around inf 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 34.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

      2. unpow2 34.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      3. associate-/l* 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      4. unpow2 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U* - U}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      5. associate-*r/ 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)} \]

      6. metadata-eval 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{\color{blue}{2}}{Om}\right)\right)\right)} \]

   6. Simplified 34.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)\right)}} \]

   7. Taylor expanded in U* around 0 11.1%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left({\ell}^{2} \cdot \left(\left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow2 11.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right) \cdot U\right)\right)\right)} \]

      2. *-commutative 11.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \left(-1 \cdot \frac{n \cdot U}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]

      3. mul-1-neg 11.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\color{blue}{\left(-\frac{n \cdot U}{{Om}^{2}}\right)} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      4. associate-/l* 11.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\color{blue}{\frac{n}{\frac{{Om}^{2}}{U}}}\right) - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      5. unpow2 11.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U}}\right) - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      6. associate-*r/ 11.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{Om \cdot Om}{U}}\right) - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      7. metadata-eval 11.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{Om \cdot Om}{U}}\right) - \frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

   9. Simplified 11.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(\left(-\frac{n}{\frac{Om \cdot Om}{U}}\right) - \frac{2}{Om}\right)\right)\right)\right)}} \]

   10. Taylor expanded in U around 0 25.3%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}\right)} \]
   11. Step-by-step derivation

       1. associate-/l* 22.4%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{Om}{U}}}\right)\right)} \]

       2. unpow2 22.4%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{Om}{U}}\right)\right)} \]

   12. Simplified 22.4%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{\ell \cdot \ell}{\frac{Om}{U}}\right)}\right)} \]

   13. Taylor expanded in n around 0 25.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
   14. Step-by-step derivation

       1. associate-/l* 24.5%

          \[\leadsto \sqrt{2 \cdot \left(-2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot U}}}\right)} \]

       2. *-commutative 24.5%

          \[\leadsto \sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{\color{blue}{U \cdot {\ell}^{2}}}}\right)} \]

       3. unpow2 24.5%

          \[\leadsto \sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}\right)} \]

   15. Simplified 24.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}} \]

   if -5800 < l < 7.19999999999999972e50

   1. Initial program 62.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 56.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 56.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 58.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 51.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 51.9%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

      2. associate-*l* 51.9%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

      3. *-commutative 51.9%

         \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5800 \lor \neg \left(\ell \leq 7.2 \cdot 10^{+50}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 18: 41.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8800 \lor \neg \left(\ell \leq 1.75 \cdot 10^{+49}\right):\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -8800.0) (not (<= l 1.75e+49)))
   (sqrt (* 2.0 (/ (* -2.0 (* n (* l (* U l)))) Om)))
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -8800.0) || !(l <= 1.75e+49)) {
		tmp = sqrt((2.0 * ((-2.0 * (n * (l * (U * l)))) / Om)));
	} else {
		tmp = pow((2.0 * (n * (U * 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 ((l <= (-8800.0d0)) .or. (.not. (l <= 1.75d+49))) then
        tmp = sqrt((2.0d0 * (((-2.0d0) * (n * (l * (u * l)))) / om)))
    else
        tmp = (2.0d0 * (n * (u * 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 ((l <= -8800.0) || !(l <= 1.75e+49)) {
		tmp = Math.sqrt((2.0 * ((-2.0 * (n * (l * (U * l)))) / Om)));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -8800.0) or not (l <= 1.75e+49):
		tmp = math.sqrt((2.0 * ((-2.0 * (n * (l * (U * l)))) / Om)))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -8800.0) || !(l <= 1.75e+49))
		tmp = sqrt(Float64(2.0 * Float64(Float64(-2.0 * Float64(n * Float64(l * Float64(U * l)))) / Om)));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -8800.0) || ~((l <= 1.75e+49)))
		tmp = sqrt((2.0 * ((-2.0 * (n * (l * (U * l)))) / Om)));
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -8800.0], N[Not[LessEqual[l, 1.75e+49]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(-2.0 * N[(n * N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8800 or 1.74999999999999987e49 < l

   1. Initial program 35.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 33.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 42.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 42.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 42.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 43.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around inf 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 34.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

      2. unpow2 34.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      3. associate-/l* 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\color{blue}{\frac{n}{\frac{{Om}^{2}}{U* - U}}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      4. unpow2 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{\color{blue}{Om \cdot Om}}{U* - U}} - 2 \cdot \frac{1}{Om}\right)\right)\right)} \]

      5. associate-*r/ 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)} \]

      6. metadata-eval 34.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{\color{blue}{2}}{Om}\right)\right)\right)} \]

   6. Simplified 34.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)\right)}} \]

   7. Taylor expanded in n around 0 25.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ 25.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{-2 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)}{Om}}} \]

      2. *-commutative 25.5%

         \[\leadsto \sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}\right)}{Om}} \]

      3. unpow2 25.5%

         \[\leadsto \sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{Om}} \]

      4. associate-*l* 29.5%

         \[\leadsto \sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \color{blue}{\left(\left(U \cdot \ell\right) \cdot \ell\right)}\right)}{Om}} \]

      5. *-commutative 29.5%

         \[\leadsto \sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \left(U \cdot \ell\right)\right)}\right)}{Om}} \]

      6. *-commutative 29.5%

         \[\leadsto \sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot U\right)}\right)\right)}{Om}} \]

   9. Simplified 29.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{-2 \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot U\right)\right)\right)}{Om}}} \]

   if -8800 < l < 1.74999999999999987e49

   1. Initial program 62.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 63.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 56.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 56.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 58.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 51.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 51.9%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

      2. associate-*l* 51.9%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

      3. *-commutative 51.9%

         \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8800 \lor \neg \left(\ell \leq 1.75 \cdot 10^{+49}\right):\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 19: 44.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -1.25e+162)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* 2.0 (* (* n U) (+ 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 <= -1.25e+162) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * ((n * U) * (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 <= (-1.25d+162)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * ((n * u) * (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 <= -1.25e+162) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -1.25e+162:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -1.25e+162)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * 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 <= -1.25e+162)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -1.25e+162], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * 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 < -1.2499999999999999e162

   1. Initial program 44.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 51.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 61.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 57.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 60.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 61.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 64.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

      2. associate-*l* 64.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

      3. *-commutative 64.3%

         \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]

   if -1.2499999999999999e162 < t

   1. Initial program 53.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 52.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 52.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 52.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 52.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 54.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 54.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 54.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 54.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 50.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 50.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 51.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 54.3%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      2. associate-*l* 54.3%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

   5. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-lft-identity 54.3%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)\right)}} \]

      2. associate-*r* 56.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

      3. *-commutative 56.9%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}}\right)\right)} \]

      4. associate-*r* 60.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \color{blue}{\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)}\right) \cdot \frac{\ell}{Om}\right)\right)} \]

   7. Simplified 60.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}} \]

   8. Taylor expanded in n around 0 43.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]
   9. Step-by-step derivation

      1. unpow2 43.7%

         \[\leadsto \sqrt{2 \cdot \left(\left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) \cdot \left(n \cdot U\right)\right)} \]

   10. Simplified 43.7%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(n \cdot U\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+162}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 20: 37.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* n (* U t))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (n * (U * t))), 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 = (2.0d0 * (n * (u * t))) ** 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((2.0 * (n * (U * t))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (n * (U * t))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * (n * (U * t))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l* 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+ 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/ 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l* 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l* 51.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow2 51.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l* 52.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified 55.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 37.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
5. Step-by-step derivation

   1. pow1/2 38.2%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

   2. associate-*l* 38.2%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]

   3. *-commutative 38.2%

      \[\leadsto {\left(2 \cdot \left(n \cdot \color{blue}{\left(U \cdot t\right)}\right)\right)}^{0.5} \]

6. Applied egg-rr 38.2%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]

7. Final simplification 38.2%

   \[\leadsto {\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5} \]

Alternative 21: 35.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* n 2.0) (* U t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((n * 2.0) * (U * t)));
}

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)))
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)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((n * 2.0) * (U * t)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(n * 2.0) * Float64(U * t)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((n * 2.0) * (U * t)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l* 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+ 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in 51.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/ 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l* 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative 55.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l* 51.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow2 51.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l* 52.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified 55.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 37.3%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

5. Final simplification 37.3%

   \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)} \]

Reproduce

herbie shell --seed 2023178 
(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*))))))