Toniolo and Linder, Equation (13)

Percentage Accurate: 49.7% → 62.9%

Time: 43.3s

Alternatives: 22

Speedup: 1.8×

Specification

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

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

Initial Program: 49.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

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

Alternative 1: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := t_3 \cdot \left(U* - U\right)\\ t_5 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_4\right)}\\ \mathbf{if}\;t_5 \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, t_1, t_3 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot t_1\right) + t_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* n (pow (/ l Om) 2.0)))
        (t_4 (* t_3 (- U* U)))
        (t_5 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_4)))))
   (if (<= t_5 2e-160)
     (* (sqrt (* 2.0 n)) (sqrt (* U (- t (fma 2.0 t_1 (* t_3 (- U U*)))))))
     (if (<= t_5 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 t_1)) t_4)))
       (sqrt
        (*
         -2.0
         (* (* U (* l l)) (* n (- (/ 2.0 Om) (* (/ n Om) (/ U* Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = n * pow((l / Om), 2.0);
	double t_4 = t_3 * (U_42_ - U);
	double t_5 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_4)));
	double tmp;
	if (t_5 <= 2e-160) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, t_1, (t_3 * (U - U_42_))))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * t_1)) + t_4)));
	} else {
		tmp = sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_4 = Float64(t_3 * Float64(U_42_ - U))
	t_5 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_4)))
	tmp = 0.0
	if (t_5 <= 2e-160)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, t_1, Float64(t_3 * Float64(U - U_42_)))))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) + t_4)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(l * l)) * Float64(n * Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 2e-160], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * t$95$1 + N[(t$95$3 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $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*))))) < 2e-160

   1. Initial program 9.7%

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

   3. Simplified 9.7%

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

      1. pow1/2 9.7%

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

      2. fma-udef 9.7%

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

      3. associate-*l/ 9.7%

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

      4. associate-*r* 9.7%

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

      5. *-commutative 9.7%

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

      6. associate--l- 9.7%

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

      7. associate-*r* 9.7%

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

      8. associate-*l* 13.5%

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

   5. Applied egg-rr 44.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. *-commutative 44.2%

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

      2. unpow1/2 44.2%

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

   7. Simplified 44.2%

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

   if 2e-160 < (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 70.4%

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

      1. associate-*l/ 75.2%

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

   3. Applied egg-rr 75.2%

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

   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/ 7.1%

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

   3. Applied egg-rr 7.1%

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

   4. Taylor expanded in n around 0 0.0%

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

      1. unpow2 0.0%

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

      2. times-frac 2.5%

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

      3. unpow2 2.5%

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

      4. associate-*r/ 7.2%

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

   6. Simplified 7.2%

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

   7. Taylor expanded in U around 0 2.9%

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

      1. associate-/l* 0.5%

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

      2. associate-*r/ 0.5%

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

      3. neg-mul-1 0.5%

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

      4. unpow2 0.5%

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

      5. times-frac 3.1%

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

      6. unpow2 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in l around inf 34.9%

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

       1. associate-*r* 34.9%

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

       2. mul-1-neg 34.9%

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

       3. +-commutative 34.9%

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

       4. sub-neg 34.9%

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

       5. unpow2 34.9%

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

       6. associate-*r/ 34.9%

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

       7. metadata-eval 34.9%

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

       8. *-commutative 34.9%

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

       9. unpow2 34.9%

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

       10. times-frac 44.1%

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

   12. Simplified 44.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 2: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 U)) (sqrt (* n t)))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l (/ l Om)))) t_1)))
       (sqrt
        (*
         -2.0
         (* (* U (* l l)) (* n (- (/ 2.0 Om) (* (/ n Om) (/ U* Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)));
	} else {
		tmp = sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)));
	} else {
		tmp = Math.sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)))
	else:
		tmp = math.sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_1)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(l * l)) * Float64(n * Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) + t_1)));
	else
		tmp = sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $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 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in t around inf 12.7%

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

      1. associate-*r* 12.7%

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

   6. Simplified 12.7%

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

      1. sqrt-prod 33.0%

         \[\leadsto \color{blue}{\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}} \]

   8. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}} \]

   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 70.2%

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

      1. associate-*l/ 75.1%

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

   3. Applied egg-rr 75.1%

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

   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/ 7.1%

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

   3. Applied egg-rr 7.1%

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

   4. Taylor expanded in n around 0 0.0%

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

      1. unpow2 0.0%

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

      2. times-frac 2.5%

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

      3. unpow2 2.5%

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

      4. associate-*r/ 7.2%

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

   6. Simplified 7.2%

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

   7. Taylor expanded in U around 0 2.9%

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

      1. associate-/l* 0.5%

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

      2. associate-*r/ 0.5%

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

      3. neg-mul-1 0.5%

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

      4. unpow2 0.5%

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

      5. times-frac 3.1%

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

      6. unpow2 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in l around inf 34.9%

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

       1. associate-*r* 34.9%

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

       2. mul-1-neg 34.9%

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

       3. +-commutative 34.9%

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

       4. sub-neg 34.9%

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

       5. unpow2 34.9%

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

       6. associate-*r/ 34.9%

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

       7. metadata-eval 34.9%

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

       8. *-commutative 34.9%

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

       9. unpow2 34.9%

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

       10. times-frac 44.1%

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

   12. Simplified 44.1%

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

4. Final simplification 64.3%

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

Alternative 3: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t_2\right) + \frac{n \cdot t_2}{Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{t_1} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))) (t_2 (* l (/ l Om))))
   (if (<= l 1.25e-253)
     (sqrt
      (* (* (* 2.0 n) U) (+ (- t (* 2.0 t_2)) (* (/ (* n t_2) Om) (- U* U)))))
     (if (<= l 7.5e-228)
       (* (sqrt t_1) (sqrt t))
       (if (<= l 7e-115)
         (sqrt
          (*
           t_1
           (+
            (+ t (* (/ (* l l) Om) -2.0))
            (* n (* (pow (/ l Om) 2.0) (- U* U))))))
         (if (<= l 3.6e+184)
           (sqrt
            (*
             2.0
             (*
              (* n U)
              (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
           (sqrt
            (*
             2.0
             (*
              U
              (*
               (* n l)
               (+
                (* (/ l Om) -2.0)
                (* (/ l Om) (/ (* n (- U* U)) Om)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = l * (l / Om);
	double tmp;
	if (l <= 1.25e-253) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_2)) + (((n * t_2) / Om) * (U_42_ - U)))));
	} else if (l <= 7.5e-228) {
		tmp = sqrt(t_1) * sqrt(t);
	} else if (l <= 7e-115) {
		tmp = sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	} else if (l <= 3.6e+184) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    t_2 = l * (l / om)
    if (l <= 1.25d-253) then
        tmp = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * t_2)) + (((n * t_2) / om) * (u_42 - u)))))
    else if (l <= 7.5d-228) then
        tmp = sqrt(t_1) * sqrt(t)
    else if (l <= 7d-115) then
        tmp = sqrt((t_1 * ((t + (((l * l) / om) * (-2.0d0))) + (n * (((l / om) ** 2.0d0) * (u_42 - u))))))
    else if (l <= 3.6d+184) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = l * (l / Om);
	double tmp;
	if (l <= 1.25e-253) {
		tmp = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_2)) + (((n * t_2) / Om) * (U_42_ - U)))));
	} else if (l <= 7.5e-228) {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	} else if (l <= 7e-115) {
		tmp = Math.sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U))))));
	} else if (l <= 3.6e+184) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	t_2 = l * (l / Om)
	tmp = 0
	if l <= 1.25e-253:
		tmp = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_2)) + (((n * t_2) / Om) * (U_42_ - U)))))
	elif l <= 7.5e-228:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	elif l <= 7e-115:
		tmp = math.sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U))))))
	elif l <= 3.6e+184:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (l <= 1.25e-253)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_2)) + Float64(Float64(Float64(n * t_2) / Om) * Float64(U_42_ - U)))));
	elseif (l <= 7.5e-228)
		tmp = Float64(sqrt(t_1) * sqrt(t));
	elseif (l <= 7e-115)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))));
	elseif (l <= 3.6e+184)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	t_2 = l * (l / Om);
	tmp = 0.0;
	if (l <= 1.25e-253)
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_2)) + (((n * t_2) / Om) * (U_42_ - U)))));
	elseif (l <= 7.5e-228)
		tmp = sqrt(t_1) * sqrt(t);
	elseif (l <= 7e-115)
		tmp = sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) + (n * (((l / Om) ^ 2.0) * (U_42_ - U))))));
	elseif (l <= 3.6e+184)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.25e-253], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n * t$95$2), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.5e-228], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7e-115], N[Sqrt[N[(t$95$1 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.6e+184], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 1.24999999999999993e-253

   1. Initial program 49.2%

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

      1. associate-*l/ 52.6%

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

   3. Applied egg-rr 52.6%

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

   4. Taylor expanded in n around 0 39.3%

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

      1. unpow2 39.3%

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

      2. times-frac 45.2%

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

      3. unpow2 45.2%

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

      4. associate-*r/ 49.7%

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

   6. Simplified 49.7%

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

      1. associate-*r/ 51.4%

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

   8. Applied egg-rr 51.4%

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

   if 1.24999999999999993e-253 < l < 7.4999999999999999e-228

   1. Initial program 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around inf 41.5%

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

      1. sqrt-prod 64.3%

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

   6. Applied egg-rr 64.3%

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

   if 7.4999999999999999e-228 < l < 7.0000000000000004e-115

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   if 7.0000000000000004e-115 < l < 3.60000000000000014e184

   1. Initial program 60.6%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in t around 0 37.4%

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

      1. distribute-lft-out 37.4%

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

      2. unpow2 37.4%

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

      3. associate-*r/ 37.4%

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

      4. *-commutative 37.4%

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

      5. associate-*l* 37.4%

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

      6. unpow2 37.4%

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

      7. unpow2 37.4%

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

   6. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. distribute-lft-out 37.4%

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

      3. associate-*l/ 37.4%

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

      4. times-frac 43.0%

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

      5. associate-*l/ 42.9%

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

      6. *-commutative 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

      2. distribute-lft-in 42.9%

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

      3. associate-*r* 48.1%

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

      4. associate-*r* 55.3%

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

      5. distribute-lft-out 57.0%

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

      6. *-commutative 57.0%

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

      7. associate-*l* 57.2%

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

      8. distribute-lft-out-- 62.3%

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

   10. Simplified 67.3%

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

   11. Taylor expanded in U around 0 56.4%

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

       1. mul-1-neg 56.4%

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

       2. associate-/l* 58.1%

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

       3. distribute-neg-frac 58.1%

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

       4. unpow2 58.1%

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

       5. times-frac 66.7%

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

   13. Simplified 66.7%

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

   14. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. *-commutative 57.1%

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

       3. distribute-rgt-neg-in 57.1%

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

       4. mul-1-neg 57.1%

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

       5. +-commutative 57.1%

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

       6. sub-neg 57.1%

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

       7. associate-*r/ 57.1%

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

       8. metadata-eval 57.1%

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

       9. *-commutative 57.1%

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

       10. unpow2 57.1%

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

       11. times-frac 71.9%

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

   16. Simplified 71.9%

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

   if 3.60000000000000014e184 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \frac{n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}{Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 4: 55.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := t - 2 \cdot t_1\\ t_3 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(t_2 + \frac{n \cdot t_1}{Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(t_2 + \left(t_1 \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))) (t_2 (- t (* 2.0 t_1))) (t_3 (* (* 2.0 n) U)))
   (if (<= l 6.5e-253)
     (sqrt (* t_3 (+ t_2 (* (/ (* n t_1) Om) (- U* U)))))
     (if (<= l 3.8e-230)
       (* (sqrt (* 2.0 (* n U))) (sqrt t))
       (if (<= l 1.3e-31)
         (sqrt (* t_3 (+ t_2 (* (* t_1 (/ n Om)) (- U* U)))))
         (if (<= l 1.35e+184)
           (sqrt
            (*
             2.0
             (*
              (* n U)
              (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
           (sqrt
            (*
             2.0
             (*
              U
              (*
               (* n l)
               (+
                (* (/ l Om) -2.0)
                (* (/ l Om) (/ (* n (- U* U)) Om)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = t - (2.0 * t_1);
	double t_3 = (2.0 * n) * U;
	double tmp;
	if (l <= 6.5e-253) {
		tmp = sqrt((t_3 * (t_2 + (((n * t_1) / Om) * (U_42_ - U)))));
	} else if (l <= 3.8e-230) {
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	} else if (l <= 1.3e-31) {
		tmp = sqrt((t_3 * (t_2 + ((t_1 * (n / Om)) * (U_42_ - U)))));
	} else if (l <= 1.35e+184) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = l * (l / om)
    t_2 = t - (2.0d0 * t_1)
    t_3 = (2.0d0 * n) * u
    if (l <= 6.5d-253) then
        tmp = sqrt((t_3 * (t_2 + (((n * t_1) / om) * (u_42 - u)))))
    else if (l <= 3.8d-230) then
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(t)
    else if (l <= 1.3d-31) then
        tmp = sqrt((t_3 * (t_2 + ((t_1 * (n / om)) * (u_42 - u)))))
    else if (l <= 1.35d+184) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = t - (2.0 * t_1);
	double t_3 = (2.0 * n) * U;
	double tmp;
	if (l <= 6.5e-253) {
		tmp = Math.sqrt((t_3 * (t_2 + (((n * t_1) / Om) * (U_42_ - U)))));
	} else if (l <= 3.8e-230) {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	} else if (l <= 1.3e-31) {
		tmp = Math.sqrt((t_3 * (t_2 + ((t_1 * (n / Om)) * (U_42_ - U)))));
	} else if (l <= 1.35e+184) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = t - (2.0 * t_1)
	t_3 = (2.0 * n) * U
	tmp = 0
	if l <= 6.5e-253:
		tmp = math.sqrt((t_3 * (t_2 + (((n * t_1) / Om) * (U_42_ - U)))))
	elif l <= 3.8e-230:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	elif l <= 1.3e-31:
		tmp = math.sqrt((t_3 * (t_2 + ((t_1 * (n / Om)) * (U_42_ - U)))))
	elif l <= 1.35e+184:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(t - Float64(2.0 * t_1))
	t_3 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l <= 6.5e-253)
		tmp = sqrt(Float64(t_3 * Float64(t_2 + Float64(Float64(Float64(n * t_1) / Om) * Float64(U_42_ - U)))));
	elseif (l <= 3.8e-230)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	elseif (l <= 1.3e-31)
		tmp = sqrt(Float64(t_3 * Float64(t_2 + Float64(Float64(t_1 * Float64(n / Om)) * Float64(U_42_ - U)))));
	elseif (l <= 1.35e+184)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = t - (2.0 * t_1);
	t_3 = (2.0 * n) * U;
	tmp = 0.0;
	if (l <= 6.5e-253)
		tmp = sqrt((t_3 * (t_2 + (((n * t_1) / Om) * (U_42_ - U)))));
	elseif (l <= 3.8e-230)
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	elseif (l <= 1.3e-31)
		tmp = sqrt((t_3 * (t_2 + ((t_1 * (n / Om)) * (U_42_ - U)))));
	elseif (l <= 1.35e+184)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l, 6.5e-253], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(N[(N[(n * t$95$1), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e-230], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e-31], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(N[(t$95$1 * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.35e+184], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 6.4999999999999998e-253

   1. Initial program 49.2%

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

      1. associate-*l/ 52.6%

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

   3. Applied egg-rr 52.6%

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

   4. Taylor expanded in n around 0 39.3%

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

      1. unpow2 39.3%

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

      2. times-frac 45.2%

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

      3. unpow2 45.2%

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

      4. associate-*r/ 49.7%

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

   6. Simplified 49.7%

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

      1. associate-*r/ 51.4%

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

   8. Applied egg-rr 51.4%

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

   if 6.4999999999999998e-253 < l < 3.7999999999999998e-230

   1. Initial program 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around inf 41.5%

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

      1. sqrt-prod 64.3%

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

   6. Applied egg-rr 64.3%

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

   if 3.7999999999999998e-230 < l < 1.29999999999999998e-31

   1. Initial program 70.7%

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

      1. associate-*l/ 70.7%

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

   3. Applied egg-rr 70.7%

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

   4. Taylor expanded in n around 0 69.5%

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

      1. unpow2 69.5%

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

      2. times-frac 72.6%

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

      3. unpow2 72.6%

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

      4. associate-*r/ 72.6%

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

   6. Simplified 72.6%

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

   if 1.29999999999999998e-31 < l < 1.35e184

   1. Initial program 49.7%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in t around 0 27.1%

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

      1. distribute-lft-out 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r/ 27.1%

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

      4. *-commutative 27.1%

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

      5. associate-*l* 27.1%

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

      6. unpow2 27.1%

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

      7. unpow2 27.1%

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

   6. Simplified 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. distribute-lft-out 27.1%

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

      3. associate-*l/ 27.1%

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

      4. times-frac 36.7%

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

      5. associate-*l/ 36.7%

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

      6. *-commutative 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. *-lft-identity 36.7%

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

      2. distribute-lft-in 36.7%

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

      3. associate-*r* 42.5%

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

      4. associate-*r* 52.6%

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

      5. distribute-lft-out 55.5%

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

      6. *-commutative 55.5%

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

      7. associate-*l* 55.5%

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

      8. distribute-lft-out-- 64.6%

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

   10. Simplified 64.6%

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

   11. Taylor expanded in U around 0 46.7%

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

       1. mul-1-neg 46.7%

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

       2. associate-/l* 46.7%

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

       3. distribute-neg-frac 46.7%

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

       4. unpow2 46.7%

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

       5. times-frac 61.7%

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

   13. Simplified 61.7%

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

   14. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. *-commutative 57.1%

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

       3. distribute-rgt-neg-in 57.1%

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

       4. mul-1-neg 57.1%

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

       5. +-commutative 57.1%

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

       6. sub-neg 57.1%

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

       7. associate-*r/ 57.1%

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

       8. metadata-eval 57.1%

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

       9. *-commutative 57.1%

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

       10. unpow2 57.1%

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

       11. times-frac 70.8%

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

   16. Simplified 70.8%

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

   if 1.35e184 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \frac{n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}{Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 5: 55.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t_1\right) + \left(t_1 \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= l 2e-34)
     (sqrt
      (* (* (* 2.0 n) U) (+ (- t (* 2.0 t_1)) (* (* t_1 (/ n Om)) (- U* U)))))
     (if (<= l 4e+184)
       (sqrt
        (*
         2.0
         (* (* n U) (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
       (sqrt
        (*
         2.0
         (*
          U
          (*
           (* n l)
           (+ (* (/ l Om) -2.0) (* (/ l Om) (/ (* n (- U* U)) Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (l <= 2e-34) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((t_1 * (n / Om)) * (U_42_ - U)))));
	} else if (l <= 4e+184) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / om)
    if (l <= 2d-34) then
        tmp = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * t_1)) + ((t_1 * (n / om)) * (u_42 - u)))))
    else if (l <= 4d+184) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (l <= 2e-34) {
		tmp = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((t_1 * (n / Om)) * (U_42_ - U)))));
	} else if (l <= 4e+184) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if l <= 2e-34:
		tmp = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((t_1 * (n / Om)) * (U_42_ - U)))))
	elif l <= 4e+184:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (l <= 2e-34)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(t_1 * Float64(n / Om)) * Float64(U_42_ - U)))));
	elseif (l <= 4e+184)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (l <= 2e-34)
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((t_1 * (n / Om)) * (U_42_ - U)))));
	elseif (l <= 4e+184)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2e-34], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4e+184], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.99999999999999986e-34

   1. Initial program 54.0%

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

      1. associate-*l/ 56.4%

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

   3. Applied egg-rr 56.4%

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

   4. Taylor expanded in n around 0 46.5%

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

      1. unpow2 46.5%

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

      2. times-frac 51.5%

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

      3. unpow2 51.5%

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

      4. associate-*r/ 54.8%

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

   6. Simplified 54.8%

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

   if 1.99999999999999986e-34 < l < 4.00000000000000007e184

   1. Initial program 49.7%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in t around 0 27.1%

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

      1. distribute-lft-out 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r/ 27.1%

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

      4. *-commutative 27.1%

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

      5. associate-*l* 27.1%

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

      6. unpow2 27.1%

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

      7. unpow2 27.1%

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

   6. Simplified 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. distribute-lft-out 27.1%

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

      3. associate-*l/ 27.1%

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

      4. times-frac 36.7%

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

      5. associate-*l/ 36.7%

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

      6. *-commutative 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. *-lft-identity 36.7%

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

      2. distribute-lft-in 36.7%

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

      3. associate-*r* 42.5%

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

      4. associate-*r* 52.6%

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

      5. distribute-lft-out 55.5%

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

      6. *-commutative 55.5%

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

      7. associate-*l* 55.5%

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

      8. distribute-lft-out-- 64.6%

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

   10. Simplified 64.6%

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

   11. Taylor expanded in U around 0 46.7%

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

       1. mul-1-neg 46.7%

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

       2. associate-/l* 46.7%

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

       3. distribute-neg-frac 46.7%

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

       4. unpow2 46.7%

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

       5. times-frac 61.7%

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

   13. Simplified 61.7%

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

   14. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. *-commutative 57.1%

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

       3. distribute-rgt-neg-in 57.1%

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

       4. mul-1-neg 57.1%

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

       5. +-commutative 57.1%

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

       6. sub-neg 57.1%

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

       7. associate-*r/ 57.1%

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

       8. metadata-eval 57.1%

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

       9. *-commutative 57.1%

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

       10. unpow2 57.1%

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

       11. times-frac 70.8%

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

   16. Simplified 70.8%

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

   if 4.00000000000000007e184 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n}{Om}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 55.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t_1\right) + \frac{n \cdot t_1}{Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= l 2e-34)
     (sqrt
      (* (* (* 2.0 n) U) (+ (- t (* 2.0 t_1)) (* (/ (* n t_1) Om) (- U* U)))))
     (if (<= l 1.25e+184)
       (sqrt
        (*
         2.0
         (* (* n U) (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
       (sqrt
        (*
         2.0
         (*
          U
          (*
           (* n l)
           (+ (* (/ l Om) -2.0) (* (/ l Om) (/ (* n (- U* U)) Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (l <= 2e-34) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + (((n * t_1) / Om) * (U_42_ - U)))));
	} else if (l <= 1.25e+184) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (l / om)
    if (l <= 2d-34) then
        tmp = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * t_1)) + (((n * t_1) / om) * (u_42 - u)))))
    else if (l <= 1.25d+184) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (l <= 2e-34) {
		tmp = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + (((n * t_1) / Om) * (U_42_ - U)))));
	} else if (l <= 1.25e+184) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if l <= 2e-34:
		tmp = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + (((n * t_1) / Om) * (U_42_ - U)))))
	elif l <= 1.25e+184:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (l <= 2e-34)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(Float64(n * t_1) / Om) * Float64(U_42_ - U)))));
	elseif (l <= 1.25e+184)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (l <= 2e-34)
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + (((n * t_1) / Om) * (U_42_ - U)))));
	elseif (l <= 1.25e+184)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2e-34], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n * t$95$1), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.25e+184], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.99999999999999986e-34

   1. Initial program 54.0%

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

      1. associate-*l/ 56.4%

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

   3. Applied egg-rr 56.4%

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

   4. Taylor expanded in n around 0 46.5%

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

      1. unpow2 46.5%

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

      2. times-frac 51.5%

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

      3. unpow2 51.5%

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

      4. associate-*r/ 54.8%

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

   6. Simplified 54.8%

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

      1. associate-*r/ 56.0%

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

   8. Applied egg-rr 56.0%

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

   if 1.99999999999999986e-34 < l < 1.25e184

   1. Initial program 49.7%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in t around 0 27.1%

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

      1. distribute-lft-out 27.1%

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

      2. unpow2 27.1%

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

      3. associate-*r/ 27.1%

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

      4. *-commutative 27.1%

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

      5. associate-*l* 27.1%

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

      6. unpow2 27.1%

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

      7. unpow2 27.1%

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

   6. Simplified 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. distribute-lft-out 27.1%

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

      3. associate-*l/ 27.1%

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

      4. times-frac 36.7%

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

      5. associate-*l/ 36.7%

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

      6. *-commutative 36.7%

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

   8. Applied egg-rr 36.7%

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

      1. *-lft-identity 36.7%

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

      2. distribute-lft-in 36.7%

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

      3. associate-*r* 42.5%

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

      4. associate-*r* 52.6%

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

      5. distribute-lft-out 55.5%

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

      6. *-commutative 55.5%

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

      7. associate-*l* 55.5%

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

      8. distribute-lft-out-- 64.6%

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

   10. Simplified 64.6%

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

   11. Taylor expanded in U around 0 46.7%

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

       1. mul-1-neg 46.7%

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

       2. associate-/l* 46.7%

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

       3. distribute-neg-frac 46.7%

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

       4. unpow2 46.7%

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

       5. times-frac 61.7%

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

   13. Simplified 61.7%

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

   14. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. *-commutative 57.1%

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

       3. distribute-rgt-neg-in 57.1%

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

       4. mul-1-neg 57.1%

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

       5. +-commutative 57.1%

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

       6. sub-neg 57.1%

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

       7. associate-*r/ 57.1%

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

       8. metadata-eval 57.1%

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

       9. *-commutative 57.1%

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

       10. unpow2 57.1%

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

       11. times-frac 70.8%

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

   16. Simplified 70.8%

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

   if 1.25e184 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \frac{n \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}{Om} \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 7: 52.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.2e-114)
   (sqrt (* 2.0 (* (* n U) (+ t (* l (* (/ U* Om) (/ (* n l) Om)))))))
   (if (<= l 3.5e+184)
     (sqrt
      (*
       2.0
       (* (* n U) (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
     (sqrt
      (*
       2.0
       (*
        U
        (*
         (* n l)
         (+ (* (/ l Om) -2.0) (* (/ l Om) (/ (* n (- U* U)) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.2e-114) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	} else if (l <= 3.5e+184) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.2d-114) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * ((u_42 / om) * ((n * l) / om)))))))
    else if (l <= 3.5d+184) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.2e-114) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	} else if (l <= 3.5e+184) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.2e-114:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))))
	elif l <= 3.5e+184:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.2e-114)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(Float64(U_42_ / Om) * Float64(Float64(n * l) / Om)))))));
	elseif (l <= 3.5e+184)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.2e-114)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	elseif (l <= 3.5e+184)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.2e-114], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.5e+184], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.2000000000000001e-114

   1. Initial program 51.0%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around 0 37.5%

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

      1. distribute-lft-out 37.5%

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

      2. unpow2 37.5%

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

      3. associate-*r/ 37.5%

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

      4. *-commutative 37.5%

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

      5. associate-*l* 37.5%

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

      6. unpow2 37.5%

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

      7. unpow2 37.5%

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

   6. Simplified 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. distribute-lft-out 37.5%

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

      3. associate-*l/ 37.5%

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

      4. times-frac 42.8%

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

      5. associate-*l/ 47.7%

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

      6. *-commutative 47.7%

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

   8. Applied egg-rr 47.7%

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

      1. *-lft-identity 47.7%

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

      2. distribute-lft-in 47.7%

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

      3. associate-*r* 49.3%

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

      4. associate-*r* 49.3%

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

      5. distribute-lft-out 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-*l* 50.1%

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

      8. distribute-lft-out-- 51.4%

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

   10. Simplified 52.0%

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

   11. Taylor expanded in U around 0 47.1%

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

       1. mul-1-neg 47.1%

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

       2. associate-/l* 47.7%

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

       3. distribute-neg-frac 47.7%

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

       4. unpow2 47.7%

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

       5. times-frac 52.5%

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

   13. Simplified 52.5%

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

   14. Taylor expanded in Om around 0 43.7%

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

       1. *-commutative 43.7%

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

       2. unpow2 43.7%

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

       3. times-frac 47.9%

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

   16. Simplified 47.9%

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

   if 1.2000000000000001e-114 < l < 3.49999999999999978e184

   1. Initial program 60.6%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in t around 0 37.4%

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

      1. distribute-lft-out 37.4%

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

      2. unpow2 37.4%

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

      3. associate-*r/ 37.4%

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

      4. *-commutative 37.4%

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

      5. associate-*l* 37.4%

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

      6. unpow2 37.4%

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

      7. unpow2 37.4%

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

   6. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. distribute-lft-out 37.4%

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

      3. associate-*l/ 37.4%

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

      4. times-frac 43.0%

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

      5. associate-*l/ 42.9%

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

      6. *-commutative 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

      2. distribute-lft-in 42.9%

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

      3. associate-*r* 48.1%

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

      4. associate-*r* 55.3%

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

      5. distribute-lft-out 57.0%

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

      6. *-commutative 57.0%

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

      7. associate-*l* 57.2%

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

      8. distribute-lft-out-- 62.3%

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

   10. Simplified 67.3%

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

   11. Taylor expanded in U around 0 56.4%

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

       1. mul-1-neg 56.4%

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

       2. associate-/l* 58.1%

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

       3. distribute-neg-frac 58.1%

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

       4. unpow2 58.1%

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

       5. times-frac 66.7%

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

   13. Simplified 66.7%

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

   14. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. *-commutative 57.1%

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

       3. distribute-rgt-neg-in 57.1%

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

       4. mul-1-neg 57.1%

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

       5. +-commutative 57.1%

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

       6. sub-neg 57.1%

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

       7. associate-*r/ 57.1%

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

       8. metadata-eval 57.1%

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

       9. *-commutative 57.1%

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

       10. unpow2 57.1%

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

       11. times-frac 71.9%

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

   16. Simplified 71.9%

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

   if 3.49999999999999978e184 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 55.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-114}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} + \frac{U*}{\frac{Om}{n} \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e-114)
   (sqrt
    (*
     2.0
     (*
      (* n U)
      (+ t (* l (+ (/ l (/ Om -2.0)) (/ U* (* (/ Om n) (/ Om l)))))))))
   (if (<= l 6.5e+182)
     (sqrt
      (*
       2.0
       (* (* n U) (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
     (sqrt
      (*
       2.0
       (*
        U
        (*
         (* n l)
         (+ (* (/ l Om) -2.0) (* (/ l Om) (/ (* n (- U* U)) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e-114) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((l / (Om / -2.0)) + (U_42_ / ((Om / n) * (Om / l)))))))));
	} else if (l <= 6.5e+182) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / 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-114) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * ((l / (om / (-2.0d0))) + (u_42 / ((om / n) * (om / l)))))))))
    else if (l <= 6.5d+182) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / 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-114) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * ((l / (Om / -2.0)) + (U_42_ / ((Om / n) * (Om / l)))))))));
	} else if (l <= 6.5e+182) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1e-114:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * ((l / (Om / -2.0)) + (U_42_ / ((Om / n) * (Om / l)))))))))
	elif l <= 6.5e+182:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e-114)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(Float64(l / Float64(Om / -2.0)) + Float64(U_42_ / Float64(Float64(Om / n) * Float64(Om / l)))))))));
	elseif (l <= 6.5e+182)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1e-114)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((l / (Om / -2.0)) + (U_42_ / ((Om / n) * (Om / l)))))))));
	elseif (l <= 6.5e+182)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e-114], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(N[(l / N[(Om / -2.0), $MachinePrecision]), $MachinePrecision] + N[(U$42$ / N[(N[(Om / n), $MachinePrecision] * N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.5e+182], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.0000000000000001e-114

   1. Initial program 51.0%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around 0 37.5%

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

      1. distribute-lft-out 37.5%

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

      2. unpow2 37.5%

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

      3. associate-*r/ 37.5%

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

      4. *-commutative 37.5%

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

      5. associate-*l* 37.5%

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

      6. unpow2 37.5%

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

      7. unpow2 37.5%

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

   6. Simplified 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. distribute-lft-out 37.5%

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

      3. associate-*l/ 37.5%

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

      4. times-frac 42.8%

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

      5. associate-*l/ 47.7%

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

      6. *-commutative 47.7%

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

   8. Applied egg-rr 47.7%

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

      1. *-lft-identity 47.7%

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

      2. distribute-lft-in 47.7%

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

      3. associate-*r* 49.3%

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

      4. associate-*r* 49.3%

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

      5. distribute-lft-out 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-*l* 50.1%

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

      8. distribute-lft-out-- 51.4%

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

   10. Simplified 52.0%

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

   11. Taylor expanded in U around 0 47.1%

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

       1. mul-1-neg 47.1%

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

       2. associate-/l* 47.7%

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

       3. distribute-neg-frac 47.7%

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

       4. unpow2 47.7%

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

       5. times-frac 52.5%

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

   13. Simplified 52.5%

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

   if 1.0000000000000001e-114 < l < 6.4999999999999998e182

   1. Initial program 60.6%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in t around 0 37.4%

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

      1. distribute-lft-out 37.4%

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

      2. unpow2 37.4%

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

      3. associate-*r/ 37.4%

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

      4. *-commutative 37.4%

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

      5. associate-*l* 37.4%

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

      6. unpow2 37.4%

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

      7. unpow2 37.4%

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

   6. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. distribute-lft-out 37.4%

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

      3. associate-*l/ 37.4%

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

      4. times-frac 43.0%

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

      5. associate-*l/ 42.9%

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

      6. *-commutative 42.9%

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

   8. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

      2. distribute-lft-in 42.9%

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

      3. associate-*r* 48.1%

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

      4. associate-*r* 55.3%

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

      5. distribute-lft-out 57.0%

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

      6. *-commutative 57.0%

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

      7. associate-*l* 57.2%

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

      8. distribute-lft-out-- 62.3%

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

   10. Simplified 67.3%

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

   11. Taylor expanded in U around 0 56.4%

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

       1. mul-1-neg 56.4%

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

       2. associate-/l* 58.1%

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

       3. distribute-neg-frac 58.1%

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

       4. unpow2 58.1%

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

       5. times-frac 66.7%

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

   13. Simplified 66.7%

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

   14. Taylor expanded in l around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. *-commutative 57.1%

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

       3. distribute-rgt-neg-in 57.1%

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

       4. mul-1-neg 57.1%

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

       5. +-commutative 57.1%

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

       6. sub-neg 57.1%

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

       7. associate-*r/ 57.1%

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

       8. metadata-eval 57.1%

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

       9. *-commutative 57.1%

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

       10. unpow2 57.1%

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

       11. times-frac 71.9%

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

   16. Simplified 71.9%

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

   if 6.4999999999999998e182 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-114}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} + \frac{U*}{\frac{Om}{n} \cdot \frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 9: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \frac{U*}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{n}}\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.25e-41)
   (sqrt
    (*
     (* (* 2.0 n) U)
     (+ (- t (* 2.0 (* l (/ l Om)))) (/ U* (* (/ Om (* l l)) (/ Om n))))))
   (if (<= l 1.65e+184)
     (sqrt
      (*
       2.0
       (* (* n U) (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om))))))))
     (sqrt
      (*
       2.0
       (*
        U
        (*
         (* n l)
         (+ (* (/ l Om) -2.0) (* (/ l Om) (/ (* n (- U* U)) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.25e-41) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * (l * (l / Om)))) + (U_42_ / ((Om / (l * l)) * (Om / n))))));
	} else if (l <= 1.65e+184) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.25d-41) then
        tmp = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * (l * (l / om)))) + (u_42 / ((om / (l * l)) * (om / n))))))
    else if (l <= 1.65d+184) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((l / om) * ((n * (u_42 - u)) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.25e-41) {
		tmp = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * (l * (l / Om)))) + (U_42_ / ((Om / (l * l)) * (Om / n))))));
	} else if (l <= 1.65e+184) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.25e-41:
		tmp = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * (l * (l / Om)))) + (U_42_ / ((Om / (l * l)) * (Om / n))))))
	elif l <= 1.65e+184:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.25e-41)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + Float64(U_42_ / Float64(Float64(Om / Float64(l * l)) * Float64(Om / n))))));
	elseif (l <= 1.65e+184)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(l / Om) * Float64(Float64(n * Float64(U_42_ - U)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.25e-41)
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * (l * (l / Om)))) + (U_42_ / ((Om / (l * l)) * (Om / n))))));
	elseif (l <= 1.65e+184)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((l / Om) * ((n * (U_42_ - U)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.25e-41], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U$42$ / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.65e+184], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.2499999999999999e-41

   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/ 55.7%

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

   3. Applied egg-rr 55.7%

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

   4. Taylor expanded in n around 0 45.7%

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

      1. unpow2 45.7%

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

      2. times-frac 50.8%

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

      3. unpow2 50.8%

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

      4. associate-*r/ 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in U around 0 45.8%

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

      1. associate-/l* 45.8%

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

      2. associate-*r/ 45.8%

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

      3. neg-mul-1 45.8%

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

      4. unpow2 45.8%

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

      5. times-frac 50.6%

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

      6. unpow2 50.6%

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

   9. Simplified 50.6%

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

   if 1.2499999999999999e-41 < l < 1.6499999999999999e184

   1. Initial program 53.9%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in t around 0 30.4%

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

      1. distribute-lft-out 30.4%

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

      2. unpow2 30.4%

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

      3. associate-*r/ 30.4%

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

      4. *-commutative 30.4%

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

      5. associate-*l* 30.4%

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

      6. unpow2 30.4%

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

      7. unpow2 30.4%

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

   6. Simplified 30.4%

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

      1. *-un-lft-identity 30.4%

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

      2. distribute-lft-out 30.4%

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

      3. associate-*l/ 30.4%

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

      4. times-frac 39.2%

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

      5. associate-*l/ 39.1%

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

      6. *-commutative 39.1%

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

   8. Applied egg-rr 39.1%

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

      1. *-lft-identity 39.1%

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

      2. distribute-lft-in 39.1%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 53.7%

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

      5. distribute-lft-out 56.5%

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

      6. *-commutative 56.5%

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

      7. associate-*l* 56.5%

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

      8. distribute-lft-out-- 64.8%

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

   10. Simplified 67.6%

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

   11. Taylor expanded in U around 0 48.3%

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

       1. mul-1-neg 48.3%

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

       2. associate-/l* 51.1%

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

       3. distribute-neg-frac 51.1%

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

       4. unpow2 51.1%

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

       5. times-frac 64.9%

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

   13. Simplified 64.9%

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

   14. Taylor expanded in l around 0 57.9%

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

       1. mul-1-neg 57.9%

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

       2. *-commutative 57.9%

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

       3. distribute-rgt-neg-in 57.9%

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

       4. mul-1-neg 57.9%

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

       5. +-commutative 57.9%

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

       6. sub-neg 57.9%

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

       7. associate-*r/ 57.9%

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

       8. metadata-eval 57.9%

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

       9. *-commutative 57.9%

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

       10. unpow2 57.9%

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

       11. times-frac 73.2%

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

   16. Simplified 73.2%

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

   if 1.6499999999999999e184 < l

   1. Initial program 18.7%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in t around 0 17.8%

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

      1. distribute-lft-out 17.8%

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

      2. unpow2 17.8%

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

      3. associate-*r/ 17.8%

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

      4. *-commutative 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

      7. unpow2 17.8%

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

   6. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. distribute-lft-out 17.8%

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

      3. associate-*l/ 17.8%

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

      4. times-frac 22.5%

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

      5. associate-*l/ 44.5%

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

      6. *-commutative 44.5%

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

   8. Applied egg-rr 44.5%

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

      1. *-lft-identity 44.5%

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

      2. distribute-lft-in 44.5%

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

      3. associate-*r* 44.5%

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

      4. associate-*r* 44.1%

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

      5. distribute-lft-out 44.1%

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

      6. *-commutative 44.1%

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

      7. associate-*l* 44.3%

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

      8. distribute-lft-out-- 45.3%

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

   10. Simplified 45.3%

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

   11. Taylor expanded in t around 0 32.1%

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

       1. associate-*r* 36.3%

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

       2. unpow2 36.3%

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

       3. times-frac 45.5%

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

   13. Simplified 45.5%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \frac{U*}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{n}}\right)}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \frac{\ell}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10: 50.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \left(n \cdot \ell\right) \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.2e+15)
   (sqrt (* 2.0 (* (* n U) (+ t (* l (* (/ U* Om) (/ (* n l) Om)))))))
   (if (<= l 1.05e+152)
     (sqrt
      (*
       -2.0
       (* (* n U) (* (* l l) (- (/ 2.0 Om) (* (/ n Om) (/ (- U* U) Om)))))))
     (sqrt
      (*
       2.0
       (*
        U
        (* (* n l) (+ (* (/ l Om) -2.0) (* (* n l) (/ U* (* Om Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+15) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	} else if (l <= 1.05e+152) {
		tmp = sqrt((-2.0 * ((n * U) * ((l * l) * ((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om)))))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((n * l) * (U_42_ / (Om * Om))))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.2d+15) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * ((u_42 / om) * ((n * l) / om)))))))
    else if (l <= 1.05d+152) then
        tmp = sqrt(((-2.0d0) * ((n * u) * ((l * l) * ((2.0d0 / om) - ((n / om) * ((u_42 - u) / om)))))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * l) * (((l / om) * (-2.0d0)) + ((n * l) * (u_42 / (om * om))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+15) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	} else if (l <= 1.05e+152) {
		tmp = Math.sqrt((-2.0 * ((n * U) * ((l * l) * ((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((n * l) * (U_42_ / (Om * Om))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.2e+15:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))))
	elif l <= 1.05e+152:
		tmp = math.sqrt((-2.0 * ((n * U) * ((l * l) * ((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om)))))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((n * l) * (U_42_ / (Om * Om))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.2e+15)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(Float64(U_42_ / Om) * Float64(Float64(n * l) / Om)))))));
	elseif (l <= 1.05e+152)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * U) * Float64(Float64(l * l) * Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(Float64(l / Om) * -2.0) + Float64(Float64(n * l) * Float64(U_42_ / Float64(Om * Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.2e+15)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	elseif (l <= 1.05e+152)
		tmp = sqrt((-2.0 * ((n * U) * ((l * l) * ((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om)))))));
	else
		tmp = sqrt((2.0 * (U * ((n * l) * (((l / Om) * -2.0) + ((n * l) * (U_42_ / (Om * Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.2e+15], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.05e+152], N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(n * l), $MachinePrecision] * N[(U$42$ / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.2e15

   1. Initial program 54.2%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around 0 39.2%

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

      1. distribute-lft-out 39.2%

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

      2. unpow2 39.2%

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

      3. associate-*r/ 39.2%

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

      4. *-commutative 39.2%

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

      5. associate-*l* 39.2%

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

      6. unpow2 39.2%

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

      7. unpow2 39.2%

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

   6. Simplified 39.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 39.2%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      2. distribute-lft-out 39.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      3. associate-*l/ 39.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

      4. times-frac 43.7%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

      5. associate-*l/ 47.8%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      6. *-commutative 47.8%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   8. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 47.8%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      2. distribute-lft-in 47.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      3. associate-*r* 49.7%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      4. associate-*r* 50.8%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

      5. distribute-lft-out 50.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      6. *-commutative 50.8%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      7. associate-*l* 51.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

      8. distribute-lft-out-- 52.6%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

   10. Simplified 54.6%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

   11. Taylor expanded in U around 0 50.2%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 50.2%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

       2. associate-/l* 51.2%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \left(-\color{blue}{\frac{U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)\right)} \]

       3. distribute-neg-frac 51.2%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)} \]

       4. unpow2 51.2%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\frac{\color{blue}{Om \cdot Om}}{\ell \cdot n}}\right)\right)\right)} \]

       5. times-frac 55.2%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\color{blue}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 55.2%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   14. Taylor expanded in Om around 0 46.3%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)} \]
   15. Step-by-step derivation

       1. *-commutative 46.3%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \frac{\color{blue}{\left(\ell \cdot n\right) \cdot U*}}{{Om}^{2}}\right)\right)} \]

       2. unpow2 46.3%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \frac{\left(\ell \cdot n\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)} \]

       3. times-frac 50.3%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\frac{\ell \cdot n}{Om} \cdot \frac{U*}{Om}\right)}\right)\right)} \]

   16. Simplified 50.3%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\frac{\ell \cdot n}{Om} \cdot \frac{U*}{Om}\right)}\right)\right)} \]

   if 6.2e15 < l < 1.0500000000000001e152

   1. Initial program 63.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 32.1%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 32.1%

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      2. *-commutative 32.1%

         \[\leadsto \sqrt{-2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      3. associate-*r/ 32.1%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{2 \cdot {\ell}^{2}}{Om}} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      4. *-commutative 32.1%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot 2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      5. associate-*r/ 32.1%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{2}{Om}} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      6. metadata-eval 32.1%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{2 \cdot 1}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      7. associate-*r/ 32.1%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left({\ell}^{2} \cdot \color{blue}{\left(2 \cdot \frac{1}{Om}\right)} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      8. associate-*r/ 32.5%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(2 \cdot \frac{1}{Om}\right) + \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      9. distribute-lft-in 43.0%

         \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}\right)} \]

      10. unpow2 43.0%

          \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. associate-*r/ 43.0%

          \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{Om}} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      12. metadata-eval 43.0%

          \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      13. *-commutative 43.0%

          \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{\color{blue}{\left(U - U*\right) \cdot n}}{{Om}^{2}}\right)\right)\right)} \]

      14. unpow2 43.0%

          \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{\left(U - U*\right) \cdot n}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      15. times-frac 63.7%

          \[\leadsto \sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \color{blue}{\frac{U - U*}{Om} \cdot \frac{n}{Om}}\right)\right)\right)} \]

   6. Simplified 63.7%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} + \frac{U - U*}{Om} \cdot \frac{n}{Om}\right)\right)\right)}} \]

   if 1.0500000000000001e152 < l

   1. Initial program 14.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 14.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 13.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 13.8%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. unpow2 13.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. associate-*r/ 13.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. *-commutative 13.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. associate-*l* 13.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. unpow2 13.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. unpow2 13.8%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

   6. Simplified 13.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 13.8%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      2. distribute-lft-out 13.8%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      3. associate-*l/ 13.8%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

      4. times-frac 17.6%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

      5. associate-*l/ 34.5%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      6. *-commutative 34.5%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   8. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 34.5%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      2. distribute-lft-in 34.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      3. associate-*r* 34.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      4. associate-*r* 37.3%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

      5. distribute-lft-out 37.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      6. *-commutative 37.3%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      7. associate-*l* 37.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

      8. distribute-lft-out-- 41.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

   10. Simplified 41.5%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

   11. Taylor expanded in U around 0 34.4%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 34.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

       2. associate-/l* 34.5%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \left(-\color{blue}{\frac{U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)\right)} \]

       3. distribute-neg-frac 34.5%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)} \]

       4. unpow2 34.5%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\frac{\color{blue}{Om \cdot Om}}{\ell \cdot n}}\right)\right)\right)} \]

       5. times-frac 41.5%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\color{blue}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 41.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   14. Taylor expanded in t around 0 28.2%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \frac{\ell}{Om} - -1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   15. Step-by-step derivation

       1. associate-*r* 31.4%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - -1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)\right)}\right)} \]

       2. mul-1-neg 31.4%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

       3. associate-*l/ 34.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - \left(-\color{blue}{\frac{U*}{{Om}^{2}} \cdot \left(\ell \cdot n\right)}\right)\right)\right)\right)} \]

       4. distribute-lft-neg-in 34.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - \color{blue}{\left(-\frac{U*}{{Om}^{2}}\right) \cdot \left(\ell \cdot n\right)}\right)\right)\right)} \]

       5. distribute-frac-neg 34.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - \color{blue}{\frac{-U*}{{Om}^{2}}} \cdot \left(\ell \cdot n\right)\right)\right)\right)} \]

       6. unpow2 34.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - \frac{-U*}{\color{blue}{Om \cdot Om}} \cdot \left(\ell \cdot n\right)\right)\right)\right)} \]

   16. Simplified 34.7%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \frac{\ell}{Om} - \frac{-U*}{Om \cdot Om} \cdot \left(\ell \cdot n\right)\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om} \cdot -2 + \left(n \cdot \ell\right) \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]

Alternative 11: 52.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{-96} \lor \neg \left(n \leq 9.4 \cdot 10^{+54}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.7e-96) (not (<= n 9.4e+54)))
   (sqrt (* 2.0 (* (* n U) (+ t (* l (* (/ U* Om) (/ (* n l) Om)))))))
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.7e-96) || !(n <= 9.4e+54)) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	} else {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-1.7d-96)) .or. (.not. (n <= 9.4d+54))) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * ((u_42 / om) * ((n * l) / om)))))))
    else
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.7e-96) || !(n <= 9.4e+54)) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.7e-96) or not (n <= 9.4e+54):
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))))
	else:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.7e-96) || !(n <= 9.4e+54))
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(Float64(U_42_ / Om) * Float64(Float64(n * l) / Om)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.7e-96) || ~((n <= 9.4e+54)))
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((U_42_ / Om) * ((n * l) / Om)))))));
	else
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.7e-96], N[Not[LessEqual[n, 9.4e+54]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.7e-96 or 9.39999999999999985e54 < n

   1. Initial program 56.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 32.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 32.4%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. unpow2 32.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. associate-*r/ 32.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. *-commutative 32.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. associate-*l* 32.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. unpow2 32.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. unpow2 32.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

   6. Simplified 32.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 32.4%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      2. distribute-lft-out 32.4%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      3. associate-*l/ 32.4%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

      4. times-frac 41.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

      5. associate-*l/ 44.4%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      6. *-commutative 44.4%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   8. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 44.4%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      2. distribute-lft-in 44.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      3. associate-*r* 48.9%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      4. associate-*r* 51.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

      5. distribute-lft-out 52.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      6. *-commutative 52.0%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      7. associate-*l* 53.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

      8. distribute-lft-out-- 57.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

   10. Simplified 63.4%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

   11. Taylor expanded in U around 0 50.8%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 50.8%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

       2. associate-/l* 52.3%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \left(-\color{blue}{\frac{U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)\right)} \]

       3. distribute-neg-frac 52.3%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)} \]

       4. unpow2 52.3%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\frac{\color{blue}{Om \cdot Om}}{\ell \cdot n}}\right)\right)\right)} \]

       5. times-frac 59.6%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\color{blue}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 59.6%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   14. Taylor expanded in Om around 0 52.4%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)} \]
   15. Step-by-step derivation

       1. *-commutative 52.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \frac{\color{blue}{\left(\ell \cdot n\right) \cdot U*}}{{Om}^{2}}\right)\right)} \]

       2. unpow2 52.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \frac{\left(\ell \cdot n\right) \cdot U*}{\color{blue}{Om \cdot Om}}\right)\right)} \]

       3. times-frac 59.6%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\frac{\ell \cdot n}{Om} \cdot \frac{U*}{Om}\right)}\right)\right)} \]

   16. Simplified 59.6%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\frac{\ell \cdot n}{Om} \cdot \frac{U*}{Om}\right)}\right)\right)} \]

   if -1.7e-96 < n < 9.39999999999999985e54

   1. Initial program 43.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 50.0%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 50.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 42.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 43.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-commutative 43.0%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. unpow2 43.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      4. associate-*r/ 49.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

      5. cancel-sign-sub-inv 49.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

      6. metadata-eval 49.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

   6. Simplified 49.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{-96} \lor \neg \left(n \leq 9.4 \cdot 10^{+54}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \end{array} \]

Alternative 12: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (* 2.0 (* (* n U) (+ t (* l (* l (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * ((n * u) * (t + (l * (l * (((n / om) * (u_42 / om)) - (2.0d0 / om))))))))
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 + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) - Float64(2.0 / Om))))))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * ((n * U) * (t + (l * (l * (((n / Om) * (U_42_ / Om)) - (2.0 / Om))))))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 49.7%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

4. Taylor expanded in t around 0 35.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. distribute-lft-out 35.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

   2. unpow2 35.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

   3. associate-*r/ 35.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. *-commutative 35.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

   5. associate-*l* 35.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

   6. unpow2 35.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

   7. unpow2 35.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

6. Simplified 35.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 35.7%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

   2. distribute-lft-out 35.7%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

   3. associate-*l/ 35.7%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

   4. times-frac 41.1%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

   5. associate-*l/ 46.3%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   6. *-commutative 46.3%

      \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

8. Applied egg-rr 46.3%

   \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-lft-identity 46.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

   2. distribute-lft-in 46.3%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

   3. associate-*r* 48.6%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   4. associate-*r* 50.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

   5. distribute-lft-out 50.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

   6. *-commutative 50.6%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   7. associate-*l* 51.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

   8. distribute-lft-out-- 53.3%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

10. Simplified 54.9%

    \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

11. Taylor expanded in U around 0 48.6%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)\right)} \]
12. Step-by-step derivation

    1. mul-1-neg 48.6%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

    2. associate-/l* 49.4%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \left(-\color{blue}{\frac{U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)\right)} \]

    3. distribute-neg-frac 49.4%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)} \]

    4. unpow2 49.4%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\frac{\color{blue}{Om \cdot Om}}{\ell \cdot n}}\right)\right)\right)} \]

    5. times-frac 55.1%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\color{blue}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

13. Simplified 55.1%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

14. Taylor expanded in l around 0 50.5%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(-1 \cdot \left(\ell \cdot \left(-1 \cdot \frac{U* \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\right)\right)} \]
15. Step-by-step derivation

    1. mul-1-neg 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(-\ell \cdot \left(-1 \cdot \frac{U* \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]

    2. *-commutative 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(-\color{blue}{\left(-1 \cdot \frac{U* \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \ell}\right)\right)\right)} \]

    3. distribute-rgt-neg-in 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\left(-1 \cdot \frac{U* \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-\ell\right)\right)}\right)\right)} \]

    4. mul-1-neg 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(\color{blue}{\left(-\frac{U* \cdot n}{{Om}^{2}}\right)} + 2 \cdot \frac{1}{Om}\right) \cdot \left(-\ell\right)\right)\right)\right)} \]

    5. +-commutative 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{Om} + \left(-\frac{U* \cdot n}{{Om}^{2}}\right)\right)} \cdot \left(-\ell\right)\right)\right)\right)} \]

    6. sub-neg 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{Om} - \frac{U* \cdot n}{{Om}^{2}}\right)} \cdot \left(-\ell\right)\right)\right)\right)} \]

    7. associate-*r/ 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(\color{blue}{\frac{2 \cdot 1}{Om}} - \frac{U* \cdot n}{{Om}^{2}}\right) \cdot \left(-\ell\right)\right)\right)\right)} \]

    8. metadata-eval 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(\frac{\color{blue}{2}}{Om} - \frac{U* \cdot n}{{Om}^{2}}\right) \cdot \left(-\ell\right)\right)\right)\right)} \]

    9. *-commutative 50.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(\frac{2}{Om} - \frac{\color{blue}{n \cdot U*}}{{Om}^{2}}\right) \cdot \left(-\ell\right)\right)\right)\right)} \]

    10. unpow2 50.5%

        \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(\frac{2}{Om} - \frac{n \cdot U*}{\color{blue}{Om \cdot Om}}\right) \cdot \left(-\ell\right)\right)\right)\right)} \]

    11. times-frac 56.3%

        \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(\frac{2}{Om} - \color{blue}{\frac{n}{Om} \cdot \frac{U*}{Om}}\right) \cdot \left(-\ell\right)\right)\right)\right)} \]

16. Simplified 56.3%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right) \cdot \left(-\ell\right)\right)}\right)\right)} \]

17. Final simplification 56.3%

    \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right)\right)\right)\right)} \]

Alternative 13: 48.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -4e+50)
   (sqrt (* 2.0 (* (* n U) (+ t (* l (* (* n l) (/ U* (* Om Om))))))))
   (if (<= n 7.2e+130)
     (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))
     (sqrt (* (* (* 2.0 n) U) (* (/ n Om) (/ U* (/ Om (* l l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -4e+50) {
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((n * l) * (U_42_ / (Om * Om))))))));
	} else if (n <= 7.2e+130) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (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 (n <= (-4d+50)) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (l * ((n * l) * (u_42 / (om * om))))))))
    else if (n <= 7.2d+130) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * ((n / om) * (u_42 / (om / (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 (n <= -4e+50) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (l * ((n * l) * (U_42_ / (Om * Om))))))));
	} else if (n <= 7.2e+130) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (l * l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -4e+50:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (l * ((n * l) * (U_42_ / (Om * Om))))))))
	elif n <= 7.2e+130:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (l * l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -4e+50)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(l * Float64(Float64(n * l) * Float64(U_42_ / Float64(Om * Om))))))));
	elseif (n <= 7.2e+130)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(n / Om) * Float64(U_42_ / Float64(Om / Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -4e+50)
		tmp = sqrt((2.0 * ((n * U) * (t + (l * ((n * l) * (U_42_ / (Om * Om))))))));
	elseif (n <= 7.2e+130)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	else
		tmp = sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -4e+50], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(l * N[(N[(n * l), $MachinePrecision] * N[(U$42$ / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 7.2e+130], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.0000000000000003e50

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 49.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 26.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 26.2%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. unpow2 26.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. associate-*r/ 26.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. *-commutative 26.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. associate-*l* 26.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. unpow2 26.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. unpow2 26.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

   6. Simplified 26.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 26.2%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      2. distribute-lft-out 26.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      3. associate-*l/ 26.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

      4. times-frac 28.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

      5. associate-*l/ 30.3%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      6. *-commutative 30.3%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   8. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 30.3%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      2. distribute-lft-in 30.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      3. associate-*r* 37.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      4. associate-*r* 37.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

      5. distribute-lft-out 39.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      6. *-commutative 39.4%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      7. associate-*l* 41.6%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

      8. distribute-lft-out-- 47.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

   10. Simplified 55.1%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

   11. Taylor expanded in U around 0 47.5%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 47.5%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

       2. associate-/l* 49.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \left(-\color{blue}{\frac{U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)\right)} \]

       3. distribute-neg-frac 49.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)} \]

       4. unpow2 49.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\frac{\color{blue}{Om \cdot Om}}{\ell \cdot n}}\right)\right)\right)} \]

       5. times-frac 51.6%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\color{blue}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 51.6%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   14. Taylor expanded in Om around 0 53.2%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*l/ 55.0%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\frac{U*}{{Om}^{2}} \cdot \left(\ell \cdot n\right)\right)}\right)\right)} \]

       2. unpow2 55.0%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{U*}{\color{blue}{Om \cdot Om}} \cdot \left(\ell \cdot n\right)\right)\right)\right)} \]

   16. Simplified 55.0%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \color{blue}{\left(\frac{U*}{Om \cdot Om} \cdot \left(\ell \cdot n\right)\right)}\right)\right)} \]

   if -4.0000000000000003e50 < n < 7.2000000000000002e130

   1. Initial program 49.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 55.4%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 55.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 43.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 45.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-commutative 45.2%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. unpow2 45.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      4. associate-*r/ 50.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

      5. cancel-sign-sub-inv 50.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

      6. metadata-eval 50.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

   6. Simplified 50.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   if 7.2000000000000002e130 < n

   1. Initial program 57.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in U* around inf 33.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-*r* 33.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{\left(U* \cdot {\ell}^{2}\right) \cdot n}}{{Om}^{2}}} \]

      2. unpow2 33.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot n}{{Om}^{2}}} \]

      3. unpow2 33.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{\color{blue}{Om \cdot Om}}} \]

   6. Simplified 33.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 33.8%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}}} \]

      2. associate-*l* 33.8%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}\right)}} \]

      3. times-frac 49.6%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)}\right)} \]

   8. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 49.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)\right)}} \]

      2. associate-*r* 49.6%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)}} \]

      3. associate-*r* 49.6%

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)} \]

      4. unpow2 49.6%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U* \cdot \color{blue}{{\ell}^{2}}}{Om} \cdot \frac{n}{Om}\right)} \]

      5. associate-/l* 49.7%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{U*}{\frac{Om}{{\ell}^{2}}}} \cdot \frac{n}{Om}\right)} \]

      6. unpow2 49.7%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U*}{\frac{Om}{\color{blue}{\ell \cdot \ell}}} \cdot \frac{n}{Om}\right)} \]

   10. Simplified 49.7%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U*}{\frac{Om}{\ell \cdot \ell}} \cdot \frac{n}{Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U*}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \end{array} \]

Alternative 14: 45.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{Om} \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 2.1e+164)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))
   (sqrt (* 2.0 (* (/ U Om) (/ (* U* (* (* l l) (* n n))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.1e+164) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = sqrt((2.0 * ((U / Om) * ((U_42_ * ((l * l) * (n * 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 (u_42 <= 2.1d+164) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    else
        tmp = sqrt((2.0d0 * ((u / om) * ((u_42 * ((l * l) * (n * 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 (U_42_ <= 2.1e+164) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = Math.sqrt((2.0 * ((U / Om) * ((U_42_ * ((l * l) * (n * n))) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 2.1e+164:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	else:
		tmp = math.sqrt((2.0 * ((U / Om) * ((U_42_ * ((l * l) * (n * n))) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 2.1e+164)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U / Om) * Float64(Float64(U_42_ * Float64(Float64(l * l) * Float64(n * n))) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 2.1e+164)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	else
		tmp = sqrt((2.0 * ((U / Om) * ((U_42_ * ((l * l) * (n * n))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 2.1e+164], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U / Om), $MachinePrecision] * N[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 2.0999999999999999e164

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 56.4%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 41.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 44.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-commutative 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. unpow2 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      4. associate-*r/ 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

      5. cancel-sign-sub-inv 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

      6. metadata-eval 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

   6. Simplified 48.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   if 2.0999999999999999e164 < U*

   1. Initial program 36.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 39.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in U* around inf 28.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 28.6%

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{\color{blue}{Om \cdot Om}}} \]

      2. times-frac 35.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\frac{U}{Om} \cdot \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}\right)}} \]

      3. unpow2 35.4%

         \[\leadsto \sqrt{2 \cdot \left(\frac{U}{Om} \cdot \frac{U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {n}^{2}\right)}{Om}\right)} \]

      4. unpow2 35.4%

         \[\leadsto \sqrt{2 \cdot \left(\frac{U}{Om} \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot n\right)}\right)}{Om}\right)} \]

   6. Simplified 35.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\frac{U}{Om} \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{Om} \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 15: 45.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot U*} \cdot \frac{Om}{n \cdot n}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 2.1e+164)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))
   (sqrt (* 2.0 (/ U (* (/ Om (* (* l l) U*)) (/ Om (* n n))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.1e+164) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = sqrt((2.0 * (U / ((Om / ((l * l) * U_42_)) * (Om / (n * n))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 2.1d+164) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    else
        tmp = sqrt((2.0d0 * (u / ((om / ((l * l) * u_42)) * (om / (n * n))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.1e+164) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = Math.sqrt((2.0 * (U / ((Om / ((l * l) * U_42_)) * (Om / (n * n))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 2.1e+164:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	else:
		tmp = math.sqrt((2.0 * (U / ((Om / ((l * l) * U_42_)) * (Om / (n * n))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 2.1e+164)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U / Float64(Float64(Om / Float64(Float64(l * l) * U_42_)) * Float64(Om / Float64(n * n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 2.1e+164)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	else
		tmp = sqrt((2.0 * (U / ((Om / ((l * l) * U_42_)) * (Om / (n * n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 2.1e+164], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U / N[(N[(Om / N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] * N[(Om / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 2.0999999999999999e164

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 56.4%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 41.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 44.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-commutative 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. unpow2 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      4. associate-*r/ 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

      5. cancel-sign-sub-inv 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

      6. metadata-eval 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

   6. Simplified 48.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   if 2.0999999999999999e164 < U*

   1. Initial program 36.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 22.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 22.0%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. unpow2 22.0%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. associate-*r/ 22.0%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. *-commutative 22.0%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. associate-*l* 22.0%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. unpow2 22.0%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. unpow2 22.0%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

   6. Simplified 22.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 22.0%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      2. distribute-lft-out 22.0%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      3. associate-*l/ 22.0%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

      4. times-frac 32.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

      5. associate-*l/ 35.6%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      6. *-commutative 35.6%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 35.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      2. distribute-lft-in 35.6%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      3. associate-*r* 33.6%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      4. associate-*r* 33.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

      5. distribute-lft-out 36.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      6. *-commutative 36.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      7. associate-*l* 36.7%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

      8. distribute-lft-out-- 40.2%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

   10. Simplified 43.3%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

   11. Taylor expanded in U* around inf 28.6%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 28.6%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U}{\frac{{Om}^{2}}{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}} \]

       2. unpow2 28.6%

          \[\leadsto \sqrt{2 \cdot \frac{U}{\frac{\color{blue}{Om \cdot Om}}{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}} \]

       3. associate-*r* 28.6%

          \[\leadsto \sqrt{2 \cdot \frac{U}{\frac{Om \cdot Om}{\color{blue}{\left(U* \cdot {\ell}^{2}\right) \cdot {n}^{2}}}}} \]

       4. times-frac 38.6%

          \[\leadsto \sqrt{2 \cdot \frac{U}{\color{blue}{\frac{Om}{U* \cdot {\ell}^{2}} \cdot \frac{Om}{{n}^{2}}}}} \]

       5. unpow2 38.6%

          \[\leadsto \sqrt{2 \cdot \frac{U}{\frac{Om}{U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}} \cdot \frac{Om}{{n}^{2}}}} \]

       6. unpow2 38.6%

          \[\leadsto \sqrt{2 \cdot \frac{U}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)} \cdot \frac{Om}{\color{blue}{n \cdot n}}}} \]

   13. Simplified 38.6%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U}{\frac{Om}{U* \cdot \left(\ell \cdot \ell\right)} \cdot \frac{Om}{n \cdot n}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{U}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot U*} \cdot \frac{Om}{n \cdot n}}}\\ \end{array} \]

Alternative 16: 46.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 2.1e+164)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))
   (sqrt (* (* (* 2.0 n) U) (* (/ n Om) (/ U* (/ Om (* l l))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.1e+164) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (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 (u_42 <= 2.1d+164) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * ((n / om) * (u_42 / (om / (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 (U_42_ <= 2.1e+164) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (l * l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 2.1e+164:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (l * l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 2.1e+164)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(n / Om) * Float64(U_42_ / Float64(Om / Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 2.1e+164)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	else
		tmp = sqrt((((2.0 * n) * U) * ((n / Om) * (U_42_ / (Om / (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 2.1e+164], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 2.0999999999999999e164

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 56.4%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 41.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 44.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-commutative 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. unpow2 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      4. associate-*r/ 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

      5. cancel-sign-sub-inv 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

      6. metadata-eval 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

   6. Simplified 48.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   if 2.0999999999999999e164 < U*

   1. Initial program 36.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in U* around inf 31.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-*r* 31.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{\left(U* \cdot {\ell}^{2}\right) \cdot n}}{{Om}^{2}}} \]

      2. unpow2 31.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot n}{{Om}^{2}}} \]

      3. unpow2 31.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{\color{blue}{Om \cdot Om}}} \]

   6. Simplified 31.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 31.8%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}}} \]

      2. associate-*l* 31.8%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}\right)}} \]

      3. times-frac 41.9%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)}\right)} \]

   8. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 41.9%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)\right)}} \]

      2. associate-*r* 41.9%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)}} \]

      3. associate-*r* 41.9%

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)} \]

      4. unpow2 41.9%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U* \cdot \color{blue}{{\ell}^{2}}}{Om} \cdot \frac{n}{Om}\right)} \]

      5. associate-/l* 42.0%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{U*}{\frac{Om}{{\ell}^{2}}}} \cdot \frac{n}{Om}\right)} \]

      6. unpow2 42.0%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U*}{\frac{Om}{\color{blue}{\ell \cdot \ell}}} \cdot \frac{n}{Om}\right)} \]

   10. Simplified 42.0%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{U*}{\frac{Om}{\ell \cdot \ell}} \cdot \frac{n}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \end{array} \]

Alternative 17: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(2 \cdot n\right) \cdot U}{Om} \cdot \left(\frac{n}{Om} \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 5e+163)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))
   (sqrt (* (/ (* (* 2.0 n) U) Om) (* (/ n Om) (* (* l l) U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 5e+163) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = sqrt(((((2.0 * n) * U) / Om) * ((n / Om) * ((l * l) * 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 (u_42 <= 5d+163) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    else
        tmp = sqrt(((((2.0d0 * n) * u) / om) * ((n / om) * ((l * l) * 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 (U_42_ <= 5e+163) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else {
		tmp = Math.sqrt(((((2.0 * n) * U) / Om) * ((n / Om) * ((l * l) * U_42_))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 5e+163:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	else:
		tmp = math.sqrt(((((2.0 * n) * U) / Om) * ((n / Om) * ((l * l) * U_42_))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 5e+163)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * n) * U) / Om) * Float64(Float64(n / Om) * Float64(Float64(l * l) * U_42_))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 5e+163)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	else
		tmp = sqrt(((((2.0 * n) * U) / Om) * ((n / Om) * ((l * l) * U_42_))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 5e+163], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 5e163

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 56.4%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in n around 0 41.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 44.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-commutative 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. unpow2 44.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      4. associate-*r/ 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

      5. cancel-sign-sub-inv 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

      6. metadata-eval 48.1%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

   6. Simplified 48.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

   if 5e163 < U*

   1. Initial program 36.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in U* around inf 31.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-*r* 31.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{\left(U* \cdot {\ell}^{2}\right) \cdot n}}{{Om}^{2}}} \]

      2. unpow2 31.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot n}{{Om}^{2}}} \]

      3. unpow2 31.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{\color{blue}{Om \cdot Om}}} \]

   6. Simplified 31.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om \cdot Om}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 31.8%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n\right)}{Om \cdot Om}}} \]

      2. *-commutative 31.8%

         \[\leadsto \sqrt{\frac{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(n \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{Om \cdot Om}} \]

   8. Applied egg-rr 31.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(n \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}} \]
   9. Step-by-step derivation

      1. times-frac 42.1%

         \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(n \cdot U\right)}{Om} \cdot \frac{n \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om}}} \]

      2. associate-*r* 42.1%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(2 \cdot n\right) \cdot U}}{Om} \cdot \frac{n \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om}} \]

      3. *-commutative 42.1%

         \[\leadsto \sqrt{\frac{\left(2 \cdot n\right) \cdot U}{Om} \cdot \frac{\color{blue}{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot n}}{Om}} \]

      4. associate-*r/ 42.1%

         \[\leadsto \sqrt{\frac{\left(2 \cdot n\right) \cdot U}{Om} \cdot \color{blue}{\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{n}{Om}\right)}} \]

   10. Simplified 42.1%

       \[\leadsto \sqrt{\color{blue}{\frac{\left(2 \cdot n\right) \cdot U}{Om} \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{n}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(2 \cdot n\right) \cdot U}{Om} \cdot \left(\frac{n}{Om} \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)\right)}\\ \end{array} \]

Alternative 18: 41.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.5e-86)
   (sqrt (* t (* 2.0 (* n U))))
   (sqrt (* 2.0 (* U (* n (+ t (* (* l (/ l Om)) -2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.5e-86) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 5.5d-86) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + ((l * (l / om)) * (-2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.5e-86) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + ((l * (l / Om)) * -2.0))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.5e-86:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + ((l * (l / Om)) * -2.0))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.5e-86)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.5e-86)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + ((l * (l / Om)) * -2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.5e-86], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5e-86

   1. Initial program 51.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 38.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}} \]

   if 5.5e-86 < l

   1. Initial program 48.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 45.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 30.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 30.9%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. unpow2 30.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. associate-*r/ 30.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(-2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. *-commutative 30.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. associate-*l* 30.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right)} - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. unpow2 30.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. unpow2 30.9%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

   6. Simplified 30.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 30.9%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      2. distribute-lft-out 30.9%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \left(\frac{\ell}{Om} \cdot -2\right) - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)}} \]

      3. associate-*l/ 30.9%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot -2}{Om}} - \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)\right)\right)} \]

      4. times-frac 36.9%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}}\right)\right)\right)} \]

      5. associate-*l/ 43.8%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      6. *-commutative 43.8%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

   8. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 43.8%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      2. distribute-lft-in 43.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right) + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right) \cdot t} + U \cdot \left(n \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      4. associate-*r* 53.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)} \]

      5. distribute-lft-out 54.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}} \]

      6. *-commutative 54.8%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)} \]

      7. associate-*l* 55.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell \cdot -2}{Om} - \color{blue}{\ell \cdot \left(\frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)\right)} \]

      8. distribute-lft-out-- 59.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{\ell \cdot \left(\frac{\ell \cdot -2}{Om} - \frac{\ell}{Om} \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right)}\right)\right)} \]

   10. Simplified 60.7%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{\frac{\ell}{Om} \cdot n}{\frac{Om}{U - U*}}\right)\right)\right)}} \]

   11. Taylor expanded in U around 0 50.4%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{-1 \cdot \frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 50.4%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\left(-\frac{U* \cdot \left(\ell \cdot n\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

       2. associate-/l* 51.8%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \left(-\color{blue}{\frac{U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)\right)} \]

       3. distribute-neg-frac 51.8%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{{Om}^{2}}{\ell \cdot n}}}\right)\right)\right)} \]

       4. unpow2 51.8%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\frac{\color{blue}{Om \cdot Om}}{\ell \cdot n}}\right)\right)\right)} \]

       5. times-frac 60.2%

          \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \frac{-U*}{\color{blue}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 60.2%

       \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \ell \cdot \left(\frac{\ell}{\frac{Om}{-2}} - \color{blue}{\frac{-U*}{\frac{Om}{\ell} \cdot \frac{Om}{n}}}\right)\right)\right)} \]

   14. Taylor expanded in n around 0 31.1%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   15. Step-by-step derivation

       1. unpow2 31.1%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)\right)} \]

       2. associate-*r/ 37.8%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

   16. Simplified 37.8%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)\right)}\\ \end{array} \]

Alternative 19: 46.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l/ 54.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Applied egg-rr 54.8%

   \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

4. Taylor expanded in n around 0 38.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 41.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   2. *-commutative 41.2%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   3. unpow2 41.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   4. associate-*r/ 44.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   5. cancel-sign-sub-inv 44.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\right)} \]

   6. metadata-eval 44.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \]

6. Simplified 44.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]

7. Final simplification 44.7%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)} \]

Alternative 20: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6e-86)
   (sqrt (* t (* 2.0 (* n U))))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6e-86) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6d-86) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6e-86) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6e-86:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6e-86)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6e-86)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6e-86], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.0000000000000002e-86

   1. Initial program 51.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 38.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}} \]

   if 6.0000000000000002e-86 < l

   1. Initial program 48.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 22.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 22.9%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   6. Simplified 22.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 27.0%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]

      2. associate-*l* 27.0%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]

   8. Applied egg-rr 27.0%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 21: 35.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* 2.0 U) (* n t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((2.0 * U) * (n * 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(((2.0d0 * u) * (n * t)))
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 * U) * (n * t)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((2.0 * U) * (n * t)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * U) * Float64(n * t)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((2.0 * U) * (n * t)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 32.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 32.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

6. Simplified 32.1%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

7. Final simplification 32.1%

   \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \]

Alternative 22: 35.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* t (* 2.0 (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((t * (2.0 * (n * U))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((t * (2.0d0 * (n * u))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((t * (2.0 * (n * U))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((t * (2.0 * (n * U))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(t * Float64(2.0 * Float64(n * U))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((t * (2.0 * (n * U))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 49.7%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

4. Taylor expanded in t around inf 33.7%

   \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}} \]

5. Final simplification 33.7%

   \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)} \]

Reproduce

herbie shell --seed 2023285 
(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*))))))