Toniolo and Linder, Equation (13)

Percentage Accurate: 49.2% → 66.1%

Time: 25.3s

Alternatives: 20

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: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_2\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(t_1, \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left(t_2 \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot t_1\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (+ -2.0 (* U* (/ n Om)))))
        (t_2 (pow (/ l Om) 2.0))
        (t_3
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n t_2) (- U* U))))))
   (if (<= t_3 0.0)
     (sqrt (* U (* (* 2.0 n) (fma t_1 (/ l Om) t))))
     (if (<= t_3 5e+300)
       (sqrt t_3)
       (if (<= t_3 INFINITY)
         (sqrt
          (*
           (* 2.0 n)
           (* U (- t (+ (* 2.0 (/ l (/ Om l))) (* n (* t_2 (- U U*))))))))
         (sqrt (* 2.0 (/ (* (* n l) (* U t_1)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (-2.0 + (U_42_ * (n / Om)));
	double t_2 = pow((l / Om), 2.0);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_2) * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * ((2.0 * n) * fma(t_1, (l / Om), t))));
	} else if (t_3 <= 5e+300) {
		tmp = sqrt(t_3);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l / (Om / l))) + (n * (t_2 * (U - U_42_))))))));
	} else {
		tmp = sqrt((2.0 * (((n * l) * (U * t_1)) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om))))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * t_2) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(t_1, Float64(l / Om), t))));
	elseif (t_3 <= 5e+300)
		tmp = sqrt(t_3);
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * Float64(l / Float64(Om / l))) + Float64(n * Float64(t_2 * Float64(U - U_42_))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * t_1)) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = 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 * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t$95$1 * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+300], N[Sqrt[t$95$3], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.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 18.0%

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

      1. associate-*l* 43.7%

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

      2. sub-neg 43.7%

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

      3. associate--l+ 43.7%

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

      4. *-commutative 43.7%

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

      5. distribute-rgt-neg-in 43.7%

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

      6. associate-*l/ 46.2%

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

      7. associate-*l* 46.2%

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

      8. *-commutative 46.2%

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

      9. *-commutative 46.2%

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

      10. associate-*l* 46.2%

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

      11. unpow2 46.2%

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

      12. associate-*l* 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in U around 0 45.9%

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

      1. div-inv 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. associate-*r/ 45.9%

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

      2. *-rgt-identity 45.9%

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

      3. associate-/l* 48.2%

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

      4. associate-/r/ 48.2%

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

   8. Simplified 48.2%

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

   9. Taylor expanded in U around 0 45.9%

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

       1. associate-*r* 45.9%

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

       2. fma-def 45.9%

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

       3. associate-*l/ 48.2%

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

       4. associate-*l/ 50.7%

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

       5. associate-*r* 55.4%

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

   11. Simplified 58.1%

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

   if 0.0 < (*.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*)))) < 5.00000000000000026e300

   1. Initial program 98.2%

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

   if 5.00000000000000026e300 < (*.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 30.0%

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

      1. associate-*l* 35.0%

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

      2. sub-neg 35.0%

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

      3. associate-+l- 35.0%

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

      4. sub-neg 35.0%

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

      5. associate-/l* 44.9%

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

      6. remove-double-neg 44.9%

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

      7. associate-*l* 44.9%

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

   3. Simplified 44.9%

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

   if +inf.0 < (*.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* 0.8%

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

      2. sub-neg 0.8%

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

      3. associate--l+ 0.8%

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

      4. *-commutative 0.8%

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

      5. distribute-rgt-neg-in 0.8%

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

      6. associate-*l/ 5.1%

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

      7. associate-*l* 5.1%

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

      8. *-commutative 5.1%

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

      9. *-commutative 5.1%

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

      10. associate-*l* 5.3%

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

      11. unpow2 5.3%

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

      12. associate-*l* 7.6%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in U around 0 42.0%

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

      1. div-inv 42.0%

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

   6. Applied egg-rr 42.0%

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

      1. associate-*r/ 42.0%

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

      2. *-rgt-identity 42.0%

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

      3. associate-/l* 42.0%

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

      4. associate-/r/ 42.0%

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

   8. Simplified 42.0%

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

   9. Taylor expanded in t around 0 58.5%

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

       1. associate-*r* 60.8%

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

       2. *-commutative 60.8%

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

       3. *-commutative 60.8%

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

       4. associate-*l/ 60.7%

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

       5. *-commutative 60.7%

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

       6. associate-*l* 62.9%

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

       7. distribute-lft-out 62.9%

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

   11. Simplified 62.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 2: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           U
           (* (* 2.0 n) (fma (* l (+ -2.0 (* U* (/ n Om)))) (/ l Om) t))))))
   (if (<= l -6.2e+241)
     (*
      (sqrt (* n (* U (+ (/ (* n U*) (* Om Om)) (/ -2.0 Om)))))
      (* (sqrt 2.0) (- l)))
     (if (<= l -2.25e-190)
       t_1
       (if (<= l 1.95e-56)
         (sqrt
          (*
           (* 2.0 n)
           (*
            U
            (-
             t
             (+
              (* 2.0 (/ l (/ Om l)))
              (* n (* (pow (/ l Om) 2.0) (- U U*))))))))
         (if (<= l 4.6e+160)
           t_1
           (*
            (* l (sqrt 2.0))
            (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((U * ((2.0 * n) * fma((l * (-2.0 + (U_42_ * (n / Om)))), (l / Om), t))));
	double tmp;
	if (l <= -6.2e+241) {
		tmp = sqrt((n * (U * (((n * U_42_) / (Om * Om)) + (-2.0 / Om))))) * (sqrt(2.0) * -l);
	} else if (l <= -2.25e-190) {
		tmp = t_1;
	} else if (l <= 1.95e-56) {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l / (Om / l))) + (n * (pow((l / Om), 2.0) * (U - U_42_))))))));
	} else if (l <= 4.6e+160) {
		tmp = t_1;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))), Float64(l / Om), t))))
	tmp = 0.0
	if (l <= -6.2e+241)
		tmp = Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Float64(Om * Om)) + Float64(-2.0 / Om))))) * Float64(sqrt(2.0) * Float64(-l)));
	elseif (l <= -2.25e-190)
		tmp = t_1;
	elseif (l <= 1.95e-56)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * Float64(l / Float64(Om / l))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U - U_42_))))))));
	elseif (l <= 4.6e+160)
		tmp = t_1;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -6.2e+241], N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.25e-190], t$95$1, If[LessEqual[l, 1.95e-56], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.6e+160], t$95$1, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.2000000000000002e241

   1. Initial program 7.8%

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

      1. associate-*l* 8.0%

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

      2. sub-neg 8.0%

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

      3. associate--l+ 8.0%

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

      4. *-commutative 8.0%

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

      5. distribute-rgt-neg-in 8.0%

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

      6. associate-*l/ 8.0%

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

      7. associate-*l* 8.0%

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

      8. *-commutative 8.0%

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

      9. *-commutative 8.0%

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

      10. associate-*l* 8.0%

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

      11. unpow2 8.0%

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

      12. associate-*l* 8.0%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in U around 0 43.5%

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

      1. div-inv 43.5%

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

   6. Applied egg-rr 43.5%

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

      1. associate-*r/ 43.5%

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

      2. *-rgt-identity 43.5%

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

      3. associate-/l* 43.5%

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

      4. associate-/r/ 43.5%

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

   8. Simplified 43.5%

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

   9. Taylor expanded in U around 0 43.5%

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

       1. associate-*r* 43.5%

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

       2. fma-def 43.5%

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

       3. associate-*l/ 43.5%

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

       4. associate-*l/ 43.5%

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

       5. associate-*r* 43.5%

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

   11. Simplified 43.5%

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

   12. Taylor expanded in l around -inf 85.5%

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

       1. mul-1-neg 85.5%

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

       2. distribute-rgt-neg-in 85.5%

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

       3. *-commutative 85.5%

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

       4. sub-neg 85.5%

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

       5. unpow2 85.5%

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

       6. associate-*r/ 85.5%

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

       7. metadata-eval 85.5%

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

       8. distribute-neg-frac 85.5%

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

       9. metadata-eval 85.5%

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

   14. Simplified 85.5%

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

   if -6.2000000000000002e241 < l < -2.2500000000000001e-190 or 1.95e-56 < l < 4.59999999999999975e160

   1. Initial program 43.8%

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

      1. associate-*l* 43.6%

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

      2. sub-neg 43.6%

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

      3. associate--l+ 43.6%

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

      4. *-commutative 43.6%

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

      5. distribute-rgt-neg-in 43.6%

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

      6. associate-*l/ 48.7%

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

      7. associate-*l* 48.7%

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

      8. *-commutative 48.7%

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

      9. *-commutative 48.7%

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

      10. associate-*l* 42.5%

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

      11. unpow2 42.5%

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

      12. associate-*l* 44.8%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in U around 0 46.7%

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

      1. div-inv 46.7%

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

   6. Applied egg-rr 46.7%

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

      1. associate-*r/ 46.7%

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

      2. *-rgt-identity 46.7%

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

      3. associate-/l* 51.9%

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

      4. associate-/r/ 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in U around 0 46.7%

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

       1. associate-*r* 46.7%

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

       2. fma-def 46.7%

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

       3. associate-*l/ 51.8%

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

       4. associate-*l/ 54.8%

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

       5. associate-*r* 56.9%

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

   11. Simplified 59.2%

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

   if -2.2500000000000001e-190 < l < 1.95e-56

   1. Initial program 73.9%

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

      1. associate-*l* 79.8%

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

      2. sub-neg 79.8%

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

      3. associate-+l- 79.8%

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

      4. sub-neg 79.8%

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

      5. associate-/l* 79.8%

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

      6. remove-double-neg 79.8%

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

      7. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

   if 4.59999999999999975e160 < l

   1. Initial program 12.9%

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

      1. associate-*l* 13.2%

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

      2. sub-neg 13.2%

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

      3. associate--l+ 13.2%

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

      4. *-commutative 13.2%

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

      5. distribute-rgt-neg-in 13.2%

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

      6. associate-*l/ 27.4%

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

      7. associate-*l* 27.4%

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

      8. *-commutative 27.4%

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

      9. *-commutative 27.4%

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

      10. associate-*l* 27.7%

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

      11. unpow2 27.7%

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

      12. associate-*l* 27.8%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in U around 0 29.7%

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

      1. div-inv 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. associate-/l* 29.7%

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

      4. associate-/r/ 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in U around 0 29.7%

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

       1. associate-*r* 29.7%

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

       2. fma-def 29.7%

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

       3. associate-*l/ 29.7%

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

       4. associate-*l/ 40.4%

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

       5. associate-*r* 36.8%

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

   11. Simplified 40.7%

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

   12. Taylor expanded in t around 0 80.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2.25 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 3: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{if}\;\ell \leq -8.4 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           U
           (* (* 2.0 n) (fma (* l (+ -2.0 (* U* (/ n Om)))) (/ l Om) t))))))
   (if (<= l -8.4e+240)
     (*
      (sqrt (* n (* U (+ (/ (* n U*) (* Om Om)) (/ -2.0 Om)))))
      (* (sqrt 2.0) (- l)))
     (if (<= l -2.3e-91)
       t_1
       (if (<= l 3.4e-39)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
         (if (<= l 8.2e+160)
           t_1
           (*
            (* l (sqrt 2.0))
            (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((U * ((2.0 * n) * fma((l * (-2.0 + (U_42_ * (n / Om)))), (l / Om), t))));
	double tmp;
	if (l <= -8.4e+240) {
		tmp = sqrt((n * (U * (((n * U_42_) / (Om * Om)) + (-2.0 / Om))))) * (sqrt(2.0) * -l);
	} else if (l <= -2.3e-91) {
		tmp = t_1;
	} else if (l <= 3.4e-39) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 8.2e+160) {
		tmp = t_1;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))), Float64(l / Om), t))))
	tmp = 0.0
	if (l <= -8.4e+240)
		tmp = Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Float64(Om * Om)) + Float64(-2.0 / Om))))) * Float64(sqrt(2.0) * Float64(-l)));
	elseif (l <= -2.3e-91)
		tmp = t_1;
	elseif (l <= 3.4e-39)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 8.2e+160)
		tmp = t_1;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -8.4e+240], N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.3e-91], t$95$1, If[LessEqual[l, 3.4e-39], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.2e+160], t$95$1, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.3999999999999996e240

   1. Initial program 7.8%

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

      1. associate-*l* 8.0%

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

      2. sub-neg 8.0%

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

      3. associate--l+ 8.0%

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

      4. *-commutative 8.0%

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

      5. distribute-rgt-neg-in 8.0%

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

      6. associate-*l/ 8.0%

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

      7. associate-*l* 8.0%

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

      8. *-commutative 8.0%

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

      9. *-commutative 8.0%

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

      10. associate-*l* 8.0%

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

      11. unpow2 8.0%

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

      12. associate-*l* 8.0%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in U around 0 43.5%

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

      1. div-inv 43.5%

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

   6. Applied egg-rr 43.5%

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

      1. associate-*r/ 43.5%

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

      2. *-rgt-identity 43.5%

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

      3. associate-/l* 43.5%

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

      4. associate-/r/ 43.5%

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

   8. Simplified 43.5%

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

   9. Taylor expanded in U around 0 43.5%

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

       1. associate-*r* 43.5%

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

       2. fma-def 43.5%

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

       3. associate-*l/ 43.5%

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

       4. associate-*l/ 43.5%

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

       5. associate-*r* 43.5%

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

   11. Simplified 43.5%

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

   12. Taylor expanded in l around -inf 85.5%

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

       1. mul-1-neg 85.5%

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

       2. distribute-rgt-neg-in 85.5%

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

       3. *-commutative 85.5%

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

       4. sub-neg 85.5%

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

       5. unpow2 85.5%

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

       6. associate-*r/ 85.5%

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

       7. metadata-eval 85.5%

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

       8. distribute-neg-frac 85.5%

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

       9. metadata-eval 85.5%

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

   14. Simplified 85.5%

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

   if -8.3999999999999996e240 < l < -2.29999999999999996e-91 or 3.3999999999999999e-39 < l < 8.19999999999999996e160

   1. Initial program 39.9%

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

      1. associate-*l* 41.5%

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

      2. sub-neg 41.5%

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

      3. associate--l+ 41.5%

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

      4. *-commutative 41.5%

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

      5. distribute-rgt-neg-in 41.5%

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

      6. associate-*l/ 48.1%

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

      7. associate-*l* 48.1%

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

      8. *-commutative 48.1%

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

      9. *-commutative 48.1%

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

      10. associate-*l* 42.2%

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

      11. unpow2 42.2%

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

      12. associate-*l* 43.2%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in U around 0 43.7%

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

      1. div-inv 43.7%

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

   6. Applied egg-rr 43.7%

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

      1. associate-*r/ 43.7%

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

      2. *-rgt-identity 43.7%

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

      3. associate-/l* 50.3%

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

      4. associate-/r/ 50.3%

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

   8. Simplified 50.3%

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

   9. Taylor expanded in U around 0 43.7%

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

       1. associate-*r* 43.7%

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

       2. fma-def 43.7%

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

       3. associate-*l/ 50.3%

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

       4. associate-*l/ 54.1%

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

       5. associate-*r* 57.0%

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

   11. Simplified 60.0%

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

   if -2.29999999999999996e-91 < l < 3.3999999999999999e-39

   1. Initial program 69.7%

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

      1. associate-*l* 72.1%

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

      2. sub-neg 72.1%

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

      3. associate--l+ 72.1%

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

      4. *-commutative 72.1%

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

      5. distribute-rgt-neg-in 72.1%

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

      6. associate-*l/ 72.1%

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

      7. associate-*l* 72.1%

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

      8. *-commutative 72.1%

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

      9. *-commutative 72.1%

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

      10. associate-*l* 60.3%

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

      11. unpow2 60.3%

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

      12. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in U around 0 72.9%

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

   if 8.19999999999999996e160 < l

   1. Initial program 12.9%

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

      1. associate-*l* 13.2%

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

      2. sub-neg 13.2%

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

      3. associate--l+ 13.2%

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

      4. *-commutative 13.2%

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

      5. distribute-rgt-neg-in 13.2%

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

      6. associate-*l/ 27.4%

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

      7. associate-*l* 27.4%

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

      8. *-commutative 27.4%

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

      9. *-commutative 27.4%

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

      10. associate-*l* 27.7%

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

      11. unpow2 27.7%

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

      12. associate-*l* 27.8%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in U around 0 29.7%

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

      1. div-inv 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. associate-/l* 29.7%

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

      4. associate-/r/ 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in U around 0 29.7%

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

       1. associate-*r* 29.7%

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

       2. fma-def 29.7%

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

       3. associate-*l/ 29.7%

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

       4. associate-*l/ 40.4%

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

       5. associate-*r* 36.8%

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

   11. Simplified 40.7%

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

   12. Taylor expanded in t around 0 80.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.4 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 4: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.65 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           U
           (* (* 2.0 n) (fma (* l (+ -2.0 (* U* (/ n Om)))) (/ l Om) t))))))
   (if (<= l -7.8e-91)
     t_1
     (if (<= l 5.9e-38)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
       (if (<= l 2.65e+160)
         t_1
         (*
          (* l (sqrt 2.0))
          (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((U * ((2.0 * n) * fma((l * (-2.0 + (U_42_ * (n / Om)))), (l / Om), t))));
	double tmp;
	if (l <= -7.8e-91) {
		tmp = t_1;
	} else if (l <= 5.9e-38) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 2.65e+160) {
		tmp = t_1;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))), Float64(l / Om), t))))
	tmp = 0.0
	if (l <= -7.8e-91)
		tmp = t_1;
	elseif (l <= 5.9e-38)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 2.65e+160)
		tmp = t_1;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -7.8e-91], t$95$1, If[LessEqual[l, 5.9e-38], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.65e+160], t$95$1, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.79999999999999987e-91 or 5.89999999999999983e-38 < l < 2.65e160

   1. Initial program 35.8%

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

      1. associate-*l* 37.3%

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

      2. sub-neg 37.3%

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

      3. associate--l+ 37.3%

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

      4. *-commutative 37.3%

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

      5. distribute-rgt-neg-in 37.3%

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

      6. associate-*l/ 43.0%

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

      7. associate-*l* 43.0%

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

      8. *-commutative 43.0%

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

      9. *-commutative 43.0%

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

      10. associate-*l* 37.9%

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

      11. unpow2 37.9%

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

      12. associate-*l* 38.8%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in U around 0 43.7%

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

      1. div-inv 43.6%

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

   6. Applied egg-rr 43.6%

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

      1. associate-*r/ 43.7%

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

      2. *-rgt-identity 43.7%

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

      3. associate-/l* 49.5%

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

      4. associate-/r/ 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in U around 0 43.7%

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

       1. associate-*r* 43.7%

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

       2. fma-def 43.7%

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

       3. associate-*l/ 49.4%

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

       4. associate-*l/ 52.7%

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

       5. associate-*r* 55.3%

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

   11. Simplified 57.9%

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

   if -7.79999999999999987e-91 < l < 5.89999999999999983e-38

   1. Initial program 69.7%

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

      1. associate-*l* 72.1%

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

      2. sub-neg 72.1%

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

      3. associate--l+ 72.1%

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

      4. *-commutative 72.1%

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

      5. distribute-rgt-neg-in 72.1%

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

      6. associate-*l/ 72.1%

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

      7. associate-*l* 72.1%

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

      8. *-commutative 72.1%

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

      9. *-commutative 72.1%

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

      10. associate-*l* 60.3%

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

      11. unpow2 60.3%

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

      12. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in U around 0 72.9%

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

   if 2.65e160 < l

   1. Initial program 12.9%

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

      1. associate-*l* 13.2%

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

      2. sub-neg 13.2%

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

      3. associate--l+ 13.2%

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

      4. *-commutative 13.2%

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

      5. distribute-rgt-neg-in 13.2%

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

      6. associate-*l/ 27.4%

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

      7. associate-*l* 27.4%

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

      8. *-commutative 27.4%

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

      9. *-commutative 27.4%

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

      10. associate-*l* 27.7%

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

      11. unpow2 27.7%

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

      12. associate-*l* 27.8%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in U around 0 29.7%

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

      1. div-inv 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. associate-*r/ 29.7%

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

      2. *-rgt-identity 29.7%

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

      3. associate-/l* 29.7%

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

      4. associate-/r/ 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in U around 0 29.7%

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

       1. associate-*r* 29.7%

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

       2. fma-def 29.7%

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

       3. associate-*l/ 29.7%

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

       4. associate-*l/ 40.4%

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

       5. associate-*r* 36.8%

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

   11. Simplified 40.7%

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

   12. Taylor expanded in t around 0 80.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.65 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right), \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 5: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+50}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} - \ell \cdot -2\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.16e+50)
   (pow
    (* 2.0 (* U (* n (- t (* (/ l Om) (- (/ n (/ (/ Om l) U)) (* l -2.0)))))))
    0.5)
   (if (<= l 7e+90)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (* (/ n Om) (* l U*)))) Om)))))
     (* (* l (sqrt 2.0)) (sqrt (/ (* n (* U (- (/ (* n U*) Om) 2.0))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.16e+50) {
		tmp = pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5);
	} else if (l <= 7e+90) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / 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 <= (-1.16d+50)) then
        tmp = (2.0d0 * (u * (n * (t - ((l / om) * ((n / ((om / l) / u)) - (l * (-2.0d0)))))))) ** 0.5d0
    else if (l <= 7d+90) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n / om) * (l * u_42)))) / om)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((n * (u * (((n * u_42) / om) - 2.0d0))) / 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 <= -1.16e+50) {
		tmp = Math.pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5);
	} else if (l <= 7e+90) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -1.16e+50:
		tmp = math.pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5)
	elif l <= 7e+90:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.16e+50)
		tmp = Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(Float64(l / Om) * Float64(Float64(n / Float64(Float64(Om / l) / U)) - Float64(l * -2.0))))))) ^ 0.5;
	elseif (l <= 7e+90)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n / Om) * Float64(l * U_42_)))) / Om)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -1.16e+50)
		tmp = (2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))) ^ 0.5;
	elseif (l <= 7e+90)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * U_42_) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.16e+50], N[Power[N[(2.0 * N[(U * N[(n * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(n / N[(N[(Om / l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 7e+90], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.16e50

   1. Initial program 23.0%

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

      1. associate-*l* 21.4%

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

      2. sub-neg 21.4%

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

      3. associate--l+ 21.4%

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

      4. *-commutative 21.4%

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

      5. distribute-rgt-neg-in 21.4%

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

      6. associate-*l/ 31.0%

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

      7. associate-*l* 31.0%

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

      8. *-commutative 31.0%

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

      9. *-commutative 31.0%

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

      10. associate-*l* 31.0%

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

      11. unpow2 31.0%

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

      12. associate-*l* 31.1%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in U* around 0 11.5%

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

      1. associate-*r* 13.4%

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

      2. +-commutative 13.4%

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

   6. Simplified 19.4%

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

      1. pow1/2 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-/r/ 49.7%

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

      4. associate-/r* 55.4%

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

   8. Applied egg-rr 55.4%

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

   if -1.16e50 < l < 6.9999999999999997e90

   1. Initial program 61.7%

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

      1. associate-*l* 64.8%

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

      2. sub-neg 64.8%

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

      3. associate--l+ 64.8%

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

      4. *-commutative 64.8%

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

      5. distribute-rgt-neg-in 64.8%

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

      6. associate-*l/ 64.8%

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

      7. associate-*l* 64.8%

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

      8. *-commutative 64.8%

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

      9. *-commutative 64.8%

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

      10. associate-*l* 53.9%

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

      11. unpow2 53.9%

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

      12. associate-*l* 55.6%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in U around 0 63.6%

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

      1. div-inv 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*r/ 63.6%

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

      2. *-rgt-identity 63.6%

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

      3. associate-/l* 67.1%

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

      4. associate-/r/ 64.8%

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

   8. Simplified 64.8%

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

   if 6.9999999999999997e90 < l

   1. Initial program 22.2%

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

      1. associate-*l* 22.5%

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

      2. sub-neg 22.5%

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

      3. associate--l+ 22.5%

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

      4. *-commutative 22.5%

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

      5. distribute-rgt-neg-in 22.5%

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

      6. associate-*l/ 37.4%

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

      7. associate-*l* 37.4%

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

      8. *-commutative 37.4%

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

      9. *-commutative 37.4%

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

      10. associate-*l* 35.0%

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

      11. unpow2 35.0%

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

      12. associate-*l* 35.2%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in U around 0 36.6%

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

      1. div-inv 36.6%

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

   6. Applied egg-rr 36.6%

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

      1. associate-*r/ 36.6%

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

      2. *-rgt-identity 36.6%

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

      3. associate-/l* 36.6%

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

      4. associate-/r/ 36.6%

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

   8. Simplified 36.6%

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

   9. Taylor expanded in U around 0 36.6%

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

       1. associate-*r* 36.6%

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

       2. fma-def 36.6%

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

       3. associate-*l/ 36.6%

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

       4. associate-*l/ 46.5%

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

       5. associate-*r* 44.0%

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

   11. Simplified 51.8%

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

   12. Taylor expanded in t around 0 74.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+50}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} - \ell \cdot -2\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 6: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-50}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} - \ell \cdot -2\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{-2 \cdot \left(n \cdot U\right)}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -2.9e-50)
   (pow
    (* 2.0 (* U (* n (- t (* (/ l Om) (- (/ n (/ (/ Om l) U)) (* l -2.0)))))))
    0.5)
   (if (<= l 4.2e+53)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (if (<= l 9.5e+231)
       (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l (+ -2.0 (* U* (/ n Om)))))))))
       (* (sqrt 2.0) (* l (sqrt (/ (* -2.0 (* n U)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -2.9e-50) {
		tmp = pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5);
	} else if (l <= 4.2e+53) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 9.5e+231) {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(((-2.0 * (n * 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 <= (-2.9d-50)) then
        tmp = (2.0d0 * (u * (n * (t - ((l / om) * ((n / ((om / l) / u)) - (l * (-2.0d0)))))))) ** 0.5d0
    else if (l <= 4.2d+53) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 9.5d+231) then
        tmp = sqrt(((2.0d0 * n) * (l / (om / (u * (l * ((-2.0d0) + (u_42 * (n / om)))))))))
    else
        tmp = sqrt(2.0d0) * (l * sqrt((((-2.0d0) * (n * 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 <= -2.9e-50) {
		tmp = Math.pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5);
	} else if (l <= 4.2e+53) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 9.5e+231) {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt(((-2.0 * (n * U)) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -2.9e-50:
		tmp = math.pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5)
	elif l <= 4.2e+53:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 9.5e+231:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt(((-2.0 * (n * U)) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -2.9e-50)
		tmp = Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(Float64(l / Om) * Float64(Float64(n / Float64(Float64(Om / l) / U)) - Float64(l * -2.0))))))) ^ 0.5;
	elseif (l <= 4.2e+53)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 9.5e+231)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(-2.0 * Float64(n * U)) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -2.9e-50)
		tmp = (2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))) ^ 0.5;
	elseif (l <= 4.2e+53)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 9.5e+231)
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	else
		tmp = sqrt(2.0) * (l * sqrt(((-2.0 * (n * U)) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -2.9e-50], N[Power[N[(2.0 * N[(U * N[(n * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(n / N[(N[(Om / l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 4.2e+53], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.5e+231], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(-2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.90000000000000008e-50

   1. Initial program 34.5%

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

      1. associate-*l* 30.6%

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

      2. sub-neg 30.6%

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

      3. associate--l+ 30.6%

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

      4. *-commutative 30.6%

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

      5. distribute-rgt-neg-in 30.6%

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

      6. associate-*l/ 37.0%

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

      7. associate-*l* 37.0%

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

      8. *-commutative 37.0%

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

      9. *-commutative 37.0%

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

      10. associate-*l* 33.0%

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

      11. unpow2 33.0%

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

      12. associate-*l* 33.0%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in U* around 0 19.9%

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

      1. associate-*r* 22.4%

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

      2. +-commutative 22.4%

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

   6. Simplified 26.5%

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

      1. pow1/2 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-/r/ 48.2%

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

      4. associate-/r* 51.9%

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

   8. Applied egg-rr 51.9%

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

   if -2.90000000000000008e-50 < l < 4.2000000000000004e53

   1. Initial program 65.1%

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

      1. associate-*l* 69.3%

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

      2. sub-neg 69.3%

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

      3. associate--l+ 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt-neg-in 69.3%

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

      6. associate-*l/ 69.3%

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

      7. associate-*l* 69.3%

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

      8. *-commutative 69.3%

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

      9. *-commutative 69.3%

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

      10. associate-*l* 58.6%

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

      11. unpow2 58.6%

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

      12. associate-*l* 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in U around 0 69.1%

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

   if 4.2000000000000004e53 < l < 9.5000000000000002e231

   1. Initial program 26.1%

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

      1. associate-*l* 31.7%

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

      2. sub-neg 31.7%

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

      3. associate--l+ 31.7%

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

      4. *-commutative 31.7%

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

      5. distribute-rgt-neg-in 31.7%

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

      6. associate-*l/ 47.3%

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

      7. associate-*l* 47.3%

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

      8. *-commutative 47.3%

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

      9. *-commutative 47.3%

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

      10. associate-*l* 41.8%

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

      11. unpow2 41.8%

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

      12. associate-*l* 44.7%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in U around 0 45.6%

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

      1. div-inv 45.6%

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

   6. Applied egg-rr 45.6%

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

      1. associate-*r/ 45.6%

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

      2. *-rgt-identity 45.6%

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

      3. associate-/l* 50.9%

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

      4. associate-/r/ 50.9%

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

   8. Simplified 50.9%

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

   9. Taylor expanded in t around 0 53.3%

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

       1. associate-/l* 56.0%

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

       2. *-commutative 56.0%

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

       3. *-commutative 56.0%

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

       4. associate-*l/ 61.2%

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

       5. *-commutative 61.2%

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

       6. associate-*l* 66.7%

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

       7. distribute-lft-out 66.7%

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

   11. Simplified 66.7%

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

   if 9.5000000000000002e231 < l

   1. Initial program 11.3%

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

      1. associate-*l* 11.3%

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

      2. sub-neg 11.3%

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

      3. associate--l+ 11.3%

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

      4. *-commutative 11.3%

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

      5. distribute-rgt-neg-in 11.3%

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

      6. associate-*l/ 11.3%

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

      7. associate-*l* 11.3%

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

      8. *-commutative 11.3%

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

      9. *-commutative 11.3%

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

      10. associate-*l* 11.8%

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

      11. unpow2 11.8%

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

      12. associate-*l* 11.8%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in l around inf 81.3%

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

      1. associate-*l* 81.7%

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

      2. sub-neg 81.7%

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

      3. unpow2 81.7%

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

      4. associate-*r/ 81.7%

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

      5. metadata-eval 81.7%

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

      6. distribute-neg-frac 81.7%

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

      7. metadata-eval 81.7%

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

   6. Simplified 81.7%

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

   7. Taylor expanded in n around 0 72.3%

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

      1. associate-*r/ 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-50}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} - \ell \cdot -2\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{-2 \cdot \left(n \cdot U\right)}{Om}}\right)\\ \end{array} \]

Alternative 7: 58.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\\ \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot t_1}{Om}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{t_1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* l (+ -2.0 (* U* (/ n Om)))))))
   (if (<= l -8.8e+80)
     (sqrt (* 2.0 (/ (* (* n l) t_1) Om)))
     (if (<= l 5.5e+56)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (* (/ n Om) (* l U*)))) Om)))))
       (sqrt (* (* 2.0 n) (/ l (/ Om t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))));
	double tmp;
	if (l <= -8.8e+80) {
		tmp = sqrt((2.0 * (((n * l) * t_1) / Om)));
	} else if (l <= 5.5e+56) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))));
	} else {
		tmp = sqrt(((2.0 * n) * (l / (Om / t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (l * ((-2.0d0) + (u_42 * (n / om))))
    if (l <= (-8.8d+80)) then
        tmp = sqrt((2.0d0 * (((n * l) * t_1) / om)))
    else if (l <= 5.5d+56) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n / om) * (l * u_42)))) / om)))))
    else
        tmp = sqrt(((2.0d0 * n) * (l / (om / t_1))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))));
	double tmp;
	if (l <= -8.8e+80) {
		tmp = Math.sqrt((2.0 * (((n * l) * t_1) / Om)));
	} else if (l <= 5.5e+56) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / t_1))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))))
	tmp = 0
	if l <= -8.8e+80:
		tmp = math.sqrt((2.0 * (((n * l) * t_1) / Om)))
	elif l <= 5.5e+56:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / t_1))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))
	tmp = 0.0
	if (l <= -8.8e+80)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * t_1) / Om)));
	elseif (l <= 5.5e+56)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n / Om) * Float64(l * U_42_)))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))));
	tmp = 0.0;
	if (l <= -8.8e+80)
		tmp = sqrt((2.0 * (((n * l) * t_1) / Om)));
	elseif (l <= 5.5e+56)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n / Om) * (l * U_42_)))) / Om)))));
	else
		tmp = sqrt(((2.0 * n) * (l / (Om / t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8.8e+80], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5.5e+56], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.80000000000000011e80

   1. Initial program 21.3%

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

      1. associate-*l* 19.5%

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

      2. sub-neg 19.5%

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

      3. associate--l+ 19.5%

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

      4. *-commutative 19.5%

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

      5. distribute-rgt-neg-in 19.5%

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

      6. associate-*l/ 30.4%

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

      7. associate-*l* 30.4%

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

      8. *-commutative 30.4%

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

      9. *-commutative 30.4%

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

      10. associate-*l* 30.4%

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

      11. unpow2 30.4%

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

      12. associate-*l* 30.5%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in U around 0 40.4%

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

      1. div-inv 40.4%

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

   6. Applied egg-rr 40.4%

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

      1. associate-*r/ 40.4%

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

      2. *-rgt-identity 40.4%

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

      3. associate-/l* 40.5%

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

      4. associate-/r/ 40.5%

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

   8. Simplified 40.5%

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

   9. Taylor expanded in t around 0 46.7%

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

       1. associate-*r* 51.1%

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

       2. *-commutative 51.1%

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

       3. *-commutative 51.1%

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

       4. associate-*l/ 51.2%

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

       5. *-commutative 51.2%

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

       6. associate-*l* 55.6%

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

       7. distribute-lft-out 55.6%

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

   11. Simplified 55.6%

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

   if -8.80000000000000011e80 < l < 5.5000000000000002e56

   1. Initial program 62.6%

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

      1. associate-*l* 64.6%

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

      2. sub-neg 64.6%

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

      3. associate--l+ 64.6%

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

      4. *-commutative 64.6%

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

      5. distribute-rgt-neg-in 64.6%

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

      6. associate-*l/ 64.6%

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

      7. associate-*l* 64.6%

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

      8. *-commutative 64.6%

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

      9. *-commutative 64.6%

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

      10. associate-*l* 53.6%

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

      11. unpow2 53.6%

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

      12. associate-*l* 54.8%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in U around 0 62.2%

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

      1. div-inv 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. associate-*r/ 62.2%

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

      2. *-rgt-identity 62.2%

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

      3. associate-/l* 66.3%

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

      4. associate-/r/ 63.9%

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

   8. Simplified 63.9%

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

   if 5.5000000000000002e56 < l

   1. Initial program 21.5%

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

      1. associate-*l* 26.0%

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

      2. sub-neg 26.0%

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

      3. associate--l+ 26.0%

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

      4. *-commutative 26.0%

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

      5. distribute-rgt-neg-in 26.0%

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

      6. associate-*l/ 38.6%

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

      7. associate-*l* 38.6%

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

      8. *-commutative 38.6%

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

      9. *-commutative 38.6%

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

      10. associate-*l* 36.4%

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

      11. unpow2 36.4%

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

      12. associate-*l* 38.7%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in U around 0 42.2%

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

      1. div-inv 42.2%

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

   6. Applied egg-rr 42.2%

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

      1. associate-*r/ 42.2%

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

      2. *-rgt-identity 42.2%

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

      3. associate-/l* 42.2%

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

      4. associate-/r/ 42.2%

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

   8. Simplified 42.2%

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

   9. Taylor expanded in t around 0 48.5%

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

       1. associate-/l* 50.6%

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

       2. *-commutative 50.6%

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

       3. *-commutative 50.6%

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

       4. associate-*l/ 52.6%

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

       5. *-commutative 52.6%

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

       6. associate-*l* 57.2%

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

       7. distribute-lft-out 57.2%

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

   11. Simplified 57.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n}{Om} \cdot \left(\ell \cdot U*\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \]

Alternative 8: 57.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-50}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} - \ell \cdot -2\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -2e-50)
   (pow
    (* 2.0 (* U (* n (- t (* (/ l Om) (- (/ n (/ (/ Om l) U)) (* l -2.0)))))))
    0.5)
   (if (<= l 9.5e+53)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (sqrt (* (* 2.0 n) (/ l (/ Om (* U (* l (+ -2.0 (* U* (/ n Om))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -2e-50) {
		tmp = pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5);
	} else if (l <= 9.5e+53) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-2d-50)) then
        tmp = (2.0d0 * (u * (n * (t - ((l / om) * ((n / ((om / l) / u)) - (l * (-2.0d0)))))))) ** 0.5d0
    else if (l <= 9.5d+53) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt(((2.0d0 * n) * (l / (om / (u * (l * ((-2.0d0) + (u_42 * (n / om)))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -2e-50) {
		tmp = Math.pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5);
	} else if (l <= 9.5e+53) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -2e-50:
		tmp = math.pow((2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))), 0.5)
	elif l <= 9.5e+53:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -2e-50)
		tmp = Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(Float64(l / Om) * Float64(Float64(n / Float64(Float64(Om / l) / U)) - Float64(l * -2.0))))))) ^ 0.5;
	elseif (l <= 9.5e+53)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -2e-50)
		tmp = (2.0 * (U * (n * (t - ((l / Om) * ((n / ((Om / l) / U)) - (l * -2.0))))))) ^ 0.5;
	elseif (l <= 9.5e+53)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(((2.0 * n) * (l / (Om / (U * (l * (-2.0 + (U_42_ * (n / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -2e-50], N[Power[N[(2.0 * N[(U * N[(n * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(n / N[(N[(Om / l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 9.5e+53], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.00000000000000002e-50

   1. Initial program 34.5%

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

      1. associate-*l* 30.6%

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

      2. sub-neg 30.6%

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

      3. associate--l+ 30.6%

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

      4. *-commutative 30.6%

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

      5. distribute-rgt-neg-in 30.6%

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

      6. associate-*l/ 37.0%

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

      7. associate-*l* 37.0%

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

      8. *-commutative 37.0%

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

      9. *-commutative 37.0%

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

      10. associate-*l* 33.0%

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

      11. unpow2 33.0%

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

      12. associate-*l* 33.0%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in U* around 0 19.9%

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

      1. associate-*r* 22.4%

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

      2. +-commutative 22.4%

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

   6. Simplified 26.5%

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

      1. pow1/2 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-/r/ 48.2%

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

      4. associate-/r* 51.9%

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

   8. Applied egg-rr 51.9%

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

   if -2.00000000000000002e-50 < l < 9.5000000000000006e53

   1. Initial program 65.1%

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

      1. associate-*l* 69.3%

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

      2. sub-neg 69.3%

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

      3. associate--l+ 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt-neg-in 69.3%

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

      6. associate-*l/ 69.3%

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

      7. associate-*l* 69.3%

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

      8. *-commutative 69.3%

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

      9. *-commutative 69.3%

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

      10. associate-*l* 58.6%

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

      11. unpow2 58.6%

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

      12. associate-*l* 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in U around 0 69.1%

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

   if 9.5000000000000006e53 < l

   1. Initial program 22.7%

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

      1. associate-*l* 27.0%

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

      2. sub-neg 27.0%

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

      3. associate--l+ 27.0%

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

      4. *-commutative 27.0%

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

      5. distribute-rgt-neg-in 27.0%

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

      6. associate-*l/ 39.0%

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

      7. associate-*l* 39.0%

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

      8. *-commutative 39.0%

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

      9. *-commutative 39.0%

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

      10. associate-*l* 34.9%

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

      11. unpow2 34.9%

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

      12. associate-*l* 37.2%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in U around 0 40.5%

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

      1. div-inv 40.5%

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

   6. Applied egg-rr 40.5%

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

      1. associate-*r/ 40.5%

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

      2. *-rgt-identity 40.5%

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

      3. associate-/l* 44.6%

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

      4. associate-/r/ 44.6%

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

   8. Simplified 44.6%

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

   9. Taylor expanded in t around 0 46.5%

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

       1. associate-/l* 48.5%

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

       2. *-commutative 48.5%

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

       3. *-commutative 48.5%

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

       4. associate-*l/ 52.5%

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

       5. *-commutative 52.5%

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

       6. associate-*l* 56.9%

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

       7. distribute-lft-out 56.9%

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

   11. Simplified 56.9%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-50}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{n}{\frac{\frac{Om}{\ell}}{U}} - \ell \cdot -2\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \]

Alternative 9: 47.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -9.6 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 3.85 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (sqrt
          (* -2.0 (/ (* (* n (* l l)) (* U (- 2.0 (/ n (/ Om U*))))) Om)))))
   (if (<= l -1.9e+44)
     t_2
     (if (<= l -9.6e-235)
       (sqrt (* (* (* 2.0 n) U) (+ t (* t_1 -2.0))))
       (if (<= l 3.85e+56)
         (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1)))))
         t_2)))))

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 = sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	double tmp;
	if (l <= -1.9e+44) {
		tmp = t_2;
	} else if (l <= -9.6e-235) {
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 3.85e+56) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = t_2;
	}
	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 = (l * l) / om
    t_2 = sqrt(((-2.0d0) * (((n * (l * l)) * (u * (2.0d0 - (n / (om / u_42))))) / om)))
    if (l <= (-1.9d+44)) then
        tmp = t_2
    else if (l <= (-9.6d-235)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (t_1 * (-2.0d0)))))
    else if (l <= 3.85d+56) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else
        tmp = t_2
    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 = Math.sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	double tmp;
	if (l <= -1.9e+44) {
		tmp = t_2;
	} else if (l <= -9.6e-235) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 3.85e+56) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	t_2 = math.sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)))
	tmp = 0
	if l <= -1.9e+44:
		tmp = t_2
	elif l <= -9.6e-235:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))))
	elif l <= 3.85e+56:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = sqrt(Float64(-2.0 * Float64(Float64(Float64(n * Float64(l * l)) * Float64(U * Float64(2.0 - Float64(n / Float64(Om / U_42_))))) / Om)))
	tmp = 0.0
	if (l <= -1.9e+44)
		tmp = t_2;
	elseif (l <= -9.6e-235)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(t_1 * -2.0))));
	elseif (l <= 3.85e+56)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * l) / Om;
	t_2 = sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	tmp = 0.0;
	if (l <= -1.9e+44)
		tmp = t_2;
	elseif (l <= -9.6e-235)
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	elseif (l <= 3.85e+56)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(-2.0 * N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.9e+44], t$95$2, If[LessEqual[l, -9.6e-235], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.85e+56], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9000000000000001e44 or 3.85e56 < l

   1. Initial program 22.0%

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

      1. associate-*l* 23.4%

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

      2. sub-neg 23.4%

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

      3. associate--l+ 23.4%

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

      4. *-commutative 23.4%

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

      5. distribute-rgt-neg-in 23.4%

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

      6. associate-*l/ 34.2%

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

      7. associate-*l* 34.2%

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

      8. *-commutative 34.2%

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

      9. *-commutative 34.2%

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

      10. associate-*l* 33.2%

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

      11. unpow2 33.2%

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

      12. associate-*l* 34.3%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in U around 0 40.6%

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

   5. Taylor expanded in l around -inf 42.4%

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

      1. associate-*r* 42.4%

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

      2. unpow2 42.4%

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

      3. *-commutative 42.4%

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

      4. mul-1-neg 42.4%

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

      5. unsub-neg 42.4%

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

      6. associate-/l* 42.4%

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

   7. Simplified 42.4%

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

   if -1.9000000000000001e44 < l < -9.60000000000000043e-235

   1. Initial program 61.5%

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

   2. Taylor expanded in Om around inf 53.5%

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

      1. unpow2 53.5%

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

   4. Simplified 53.5%

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

   if -9.60000000000000043e-235 < l < 3.85e56

   1. Initial program 65.7%

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

      1. associate-*l* 73.6%

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

      2. sub-neg 73.6%

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

      3. associate-+l- 73.6%

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

      4. sub-neg 73.6%

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

      5. associate-/l* 73.6%

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

      6. remove-double-neg 73.6%

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

      7. associate-*l* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in Om around inf 66.0%

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

      1. unpow2 66.0%

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

   6. Simplified 66.0%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -9.6 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 3.85 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \end{array} \]

Alternative 10: 52.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (sqrt
          (* 2.0 (/ (* (* n l) (* U (* l (+ -2.0 (* U* (/ n Om)))))) Om)))))
   (if (<= l -8e+38)
     t_2
     (if (<= l -2.3e-228)
       (sqrt (* (* (* 2.0 n) U) (+ t (* t_1 -2.0))))
       (if (<= l 5e+56) (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1))))) t_2)))))

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 = sqrt((2.0 * (((n * l) * (U * (l * (-2.0 + (U_42_ * (n / Om)))))) / Om)));
	double tmp;
	if (l <= -8e+38) {
		tmp = t_2;
	} else if (l <= -2.3e-228) {
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 5e+56) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = t_2;
	}
	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 = (l * l) / om
    t_2 = sqrt((2.0d0 * (((n * l) * (u * (l * ((-2.0d0) + (u_42 * (n / om)))))) / om)))
    if (l <= (-8d+38)) then
        tmp = t_2
    else if (l <= (-2.3d-228)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (t_1 * (-2.0d0)))))
    else if (l <= 5d+56) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else
        tmp = t_2
    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 = Math.sqrt((2.0 * (((n * l) * (U * (l * (-2.0 + (U_42_ * (n / Om)))))) / Om)));
	double tmp;
	if (l <= -8e+38) {
		tmp = t_2;
	} else if (l <= -2.3e-228) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 5e+56) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	t_2 = math.sqrt((2.0 * (((n * l) * (U * (l * (-2.0 + (U_42_ * (n / Om)))))) / Om)))
	tmp = 0
	if l <= -8e+38:
		tmp = t_2
	elif l <= -2.3e-228:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))))
	elif l <= 5e+56:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))) / Om)))
	tmp = 0.0
	if (l <= -8e+38)
		tmp = t_2;
	elseif (l <= -2.3e-228)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(t_1 * -2.0))));
	elseif (l <= 5e+56)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * l) / Om;
	t_2 = sqrt((2.0 * (((n * l) * (U * (l * (-2.0 + (U_42_ * (n / Om)))))) / Om)));
	tmp = 0.0;
	if (l <= -8e+38)
		tmp = t_2;
	elseif (l <= -2.3e-228)
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	elseif (l <= 5e+56)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -8e+38], t$95$2, If[LessEqual[l, -2.3e-228], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 5e+56], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.99999999999999982e38 or 5.00000000000000024e56 < l

   1. Initial program 22.6%

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

      1. associate-*l* 23.9%

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

      2. sub-neg 23.9%

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

      3. associate--l+ 23.9%

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

      4. *-commutative 23.9%

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

      5. distribute-rgt-neg-in 23.9%

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

      6. associate-*l/ 34.5%

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

      7. associate-*l* 34.5%

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

      8. *-commutative 34.5%

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

      9. *-commutative 34.5%

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

      10. associate-*l* 32.6%

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

      11. unpow2 32.6%

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

      12. associate-*l* 33.7%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in U around 0 39.9%

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

      1. div-inv 39.9%

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

   6. Applied egg-rr 39.9%

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

      1. associate-*r/ 39.9%

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

      2. *-rgt-identity 39.9%

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

      3. associate-/l* 41.8%

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

      4. associate-/r/ 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in t around 0 46.1%

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

       1. associate-*r* 49.1%

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

       2. *-commutative 49.1%

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

       3. *-commutative 49.1%

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

       4. associate-*l/ 49.2%

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

       5. *-commutative 49.2%

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

       6. associate-*l* 53.3%

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

       7. distribute-lft-out 53.3%

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

   11. Simplified 53.3%

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

   if -7.99999999999999982e38 < l < -2.2999999999999999e-228

   1. Initial program 61.9%

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

   2. Taylor expanded in Om around inf 53.7%

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

      1. unpow2 53.7%

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

   4. Simplified 53.7%

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

   if -2.2999999999999999e-228 < l < 5.00000000000000024e56

   1. Initial program 65.7%

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

      1. associate-*l* 73.6%

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

      2. sub-neg 73.6%

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

      3. associate-+l- 73.6%

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

      4. sub-neg 73.6%

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

      5. associate-/l* 73.6%

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

      6. remove-double-neg 73.6%

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

      7. associate-*l* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in Om around inf 66.0%

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

      1. unpow2 66.0%

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

   6. Simplified 66.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -2.3 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 11: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot t_1}{Om}}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_2 \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{t_1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* l (+ -2.0 (* U* (/ n Om)))))) (t_2 (/ (* l l) Om)))
   (if (<= l -1e+39)
     (sqrt (* 2.0 (/ (* (* n l) t_1) Om)))
     (if (<= l -7e-229)
       (sqrt (* (* (* 2.0 n) U) (+ t (* t_2 -2.0))))
       (if (<= l 1.6e+53)
         (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_2)))))
         (sqrt (* (* 2.0 n) (/ l (/ Om t_1)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))));
	double t_2 = (l * l) / Om;
	double tmp;
	if (l <= -1e+39) {
		tmp = sqrt((2.0 * (((n * l) * t_1) / Om)));
	} else if (l <= -7e-229) {
		tmp = sqrt((((2.0 * n) * U) * (t + (t_2 * -2.0))));
	} else if (l <= 1.6e+53) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_2)))));
	} else {
		tmp = sqrt(((2.0 * n) * (l / (Om / t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = u * (l * ((-2.0d0) + (u_42 * (n / om))))
    t_2 = (l * l) / om
    if (l <= (-1d+39)) then
        tmp = sqrt((2.0d0 * (((n * l) * t_1) / om)))
    else if (l <= (-7d-229)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (t_2 * (-2.0d0)))))
    else if (l <= 1.6d+53) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_2)))))
    else
        tmp = sqrt(((2.0d0 * n) * (l / (om / t_1))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))));
	double t_2 = (l * l) / Om;
	double tmp;
	if (l <= -1e+39) {
		tmp = Math.sqrt((2.0 * (((n * l) * t_1) / Om)));
	} else if (l <= -7e-229) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (t_2 * -2.0))));
	} else if (l <= 1.6e+53) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_2)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (l / (Om / t_1))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))))
	t_2 = (l * l) / Om
	tmp = 0
	if l <= -1e+39:
		tmp = math.sqrt((2.0 * (((n * l) * t_1) / Om)))
	elif l <= -7e-229:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (t_2 * -2.0))))
	elif l <= 1.6e+53:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_2)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (l / (Om / t_1))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(l * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))
	t_2 = Float64(Float64(l * l) / Om)
	tmp = 0.0
	if (l <= -1e+39)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * t_1) / Om)));
	elseif (l <= -7e-229)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(t_2 * -2.0))));
	elseif (l <= 1.6e+53)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_2)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(l / Float64(Om / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (l * (-2.0 + (U_42_ * (n / Om))));
	t_2 = (l * l) / Om;
	tmp = 0.0;
	if (l <= -1e+39)
		tmp = sqrt((2.0 * (((n * l) * t_1) / Om)));
	elseif (l <= -7e-229)
		tmp = sqrt((((2.0 * n) * U) * (t + (t_2 * -2.0))));
	elseif (l <= 1.6e+53)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_2)))));
	else
		tmp = sqrt(((2.0 * n) * (l / (Om / t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(l * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, -1e+39], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -7e-229], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(t$95$2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.6e+53], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(l / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.9999999999999994e38

   1. Initial program 23.5%

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

      1. associate-*l* 22.1%

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

      2. sub-neg 22.1%

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

      3. associate--l+ 22.1%

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

      4. *-commutative 22.1%

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

      5. distribute-rgt-neg-in 22.1%

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

      6. associate-*l/ 31.1%

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

      7. associate-*l* 31.1%

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

      8. *-commutative 31.1%

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

      9. *-commutative 31.1%

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

      10. associate-*l* 29.3%

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

      11. unpow2 29.3%

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

      12. associate-*l* 29.4%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in U around 0 37.9%

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

      1. div-inv 37.9%

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

   6. Applied egg-rr 37.9%

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

      1. associate-*r/ 37.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot 1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

      2. *-rgt-identity 37.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{\color{blue}{n \cdot \left(\ell \cdot U*\right)}}{Om}\right)}{Om}\right) \cdot U\right)} \]

      3. associate-/l* 41.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{\frac{Om}{\ell \cdot U*}}}\right)}{Om}\right) \cdot U\right)} \]

      4. associate-/r/ 41.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   8. Simplified 41.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   9. Taylor expanded in t around 0 45.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \left(\ell \cdot \left(\left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot U\right)\right)}{Om}}} \]
   10. Step-by-step derivation

       1. associate-*r* 48.6%

          \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(\left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot U\right)}}{Om}} \]

       2. *-commutative 48.6%

          \[\leadsto \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}}{Om}} \]

       3. *-commutative 48.6%

          \[\leadsto \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}{Om}} \]

       4. associate-*l/ 48.8%

          \[\leadsto \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot -2 + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)\right)}{Om}} \]

       5. *-commutative 48.8%

          \[\leadsto \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot -2 + \color{blue}{\left(\ell \cdot U*\right) \cdot \frac{n}{Om}}\right)\right)}{Om}} \]

       6. associate-*l* 52.5%

          \[\leadsto \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot -2 + \color{blue}{\ell \cdot \left(U* \cdot \frac{n}{Om}\right)}\right)\right)}{Om}} \]

       7. distribute-lft-out 52.5%

          \[\leadsto \sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}\right)}{Om}} \]

   11. Simplified 52.5%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}} \]

   if -9.9999999999999994e38 < l < -7.0000000000000007e-229

   1. Initial program 61.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 53.7%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

   4. Simplified 53.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   if -7.0000000000000007e-229 < l < 1.6e53

   1. Initial program 66.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 74.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate-+l- 74.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

      4. sub-neg 74.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      5. associate-/l* 74.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

      6. remove-double-neg 74.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

      7. associate-*l* 74.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 66.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 66.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   6. Simplified 66.3%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right)} \]

   if 1.6e53 < l

   1. Initial program 22.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 27.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 27.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 27.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 27.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 27.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 39.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 39.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 39.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 39.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 34.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 34.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 37.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 50.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U around 0 40.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. div-inv 40.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot \frac{1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

   6. Applied egg-rr 40.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot \frac{1}{Om}}\right)}{Om}\right) \cdot U\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 40.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot 1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

      2. *-rgt-identity 40.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{\color{blue}{n \cdot \left(\ell \cdot U*\right)}}{Om}\right)}{Om}\right) \cdot U\right)} \]

      3. associate-/l* 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{\frac{Om}{\ell \cdot U*}}}\right)}{Om}\right) \cdot U\right)} \]

      4. associate-/r/ 44.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   8. Simplified 44.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   9. Taylor expanded in t around 0 46.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{\ell \cdot \left(\left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot U\right)}{Om}}} \]
   10. Step-by-step derivation

       1. associate-/l* 48.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{\ell}{\frac{Om}{\left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot U}}}} \]

       2. *-commutative 48.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{\color{blue}{U \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}}}} \]

       3. *-commutative 48.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\color{blue}{\ell \cdot -2} + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}}} \]

       4. associate-*l/ 52.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot -2 + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}}} \]

       5. *-commutative 52.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot -2 + \color{blue}{\left(\ell \cdot U*\right) \cdot \frac{n}{Om}}\right)}}} \]

       6. associate-*l* 56.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot -2 + \color{blue}{\ell \cdot \left(U* \cdot \frac{n}{Om}\right)}\right)}}} \]

       7. distribute-lft-out 56.9%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \color{blue}{\left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}} \]

   11. Simplified 56.9%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;\ell \leq -7 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\ell}{\frac{Om}{U \cdot \left(\ell \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \]

Alternative 12: 44.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4.3 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (sqrt (* 2.0 (* (/ (* (* n l) (* n l)) Om) (/ (* U U*) Om))))))
   (if (<= l -3.3e+105)
     t_2
     (if (<= l -4.3e-235)
       (sqrt (* (* (* 2.0 n) U) (+ t (* t_1 -2.0))))
       (if (<= l 3.7e+92) (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1))))) t_2)))))

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 = sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	double tmp;
	if (l <= -3.3e+105) {
		tmp = t_2;
	} else if (l <= -4.3e-235) {
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 3.7e+92) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = t_2;
	}
	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 = (l * l) / om
    t_2 = sqrt((2.0d0 * ((((n * l) * (n * l)) / om) * ((u * u_42) / om))))
    if (l <= (-3.3d+105)) then
        tmp = t_2
    else if (l <= (-4.3d-235)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (t_1 * (-2.0d0)))))
    else if (l <= 3.7d+92) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else
        tmp = t_2
    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 = Math.sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	double tmp;
	if (l <= -3.3e+105) {
		tmp = t_2;
	} else if (l <= -4.3e-235) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 3.7e+92) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	t_2 = math.sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))))
	tmp = 0
	if l <= -3.3e+105:
		tmp = t_2
	elif l <= -4.3e-235:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))))
	elif l <= 3.7e+92:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(n * l) * Float64(n * l)) / Om) * Float64(Float64(U * U_42_) / Om))))
	tmp = 0.0
	if (l <= -3.3e+105)
		tmp = t_2;
	elseif (l <= -4.3e-235)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(t_1 * -2.0))));
	elseif (l <= 3.7e+92)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * l) / Om;
	t_2 = sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	tmp = 0.0;
	if (l <= -3.3e+105)
		tmp = t_2;
	elseif (l <= -4.3e-235)
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	elseif (l <= 3.7e+92)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.3e+105], t$95$2, If[LessEqual[l, -4.3e-235], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.7e+92], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.29999999999999997e105 or 3.69999999999999999e92 < l

   1. Initial program 18.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 20.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 20.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 20.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 20.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 20.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 33.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 33.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 33.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 33.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 32.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 32.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 32.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 50.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U around 0 38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. div-inv 38.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot \frac{1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

   6. Applied egg-rr 38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot \frac{1}{Om}}\right)}{Om}\right) \cdot U\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 38.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot 1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

      2. *-rgt-identity 38.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{\color{blue}{n \cdot \left(\ell \cdot U*\right)}}{Om}\right)}{Om}\right) \cdot U\right)} \]

      3. associate-/l* 38.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{\frac{Om}{\ell \cdot U*}}}\right)}{Om}\right) \cdot U\right)} \]

      4. associate-/r/ 38.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   8. Simplified 38.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   9. Taylor expanded in n around inf 29.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-*r* 29.9%

          \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(U \cdot U*\right)}}{{Om}^{2}}} \]

       2. *-commutative 29.9%

          \[\leadsto \sqrt{2 \cdot \frac{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \color{blue}{\left(U* \cdot U\right)}}{{Om}^{2}}} \]

       3. unpow2 29.9%

          \[\leadsto \sqrt{2 \cdot \frac{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(U* \cdot U\right)}{\color{blue}{Om \cdot Om}}} \]

       4. times-frac 32.6%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\frac{{n}^{2} \cdot {\ell}^{2}}{Om} \cdot \frac{U* \cdot U}{Om}\right)}} \]

       5. unpow2 32.6%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\color{blue}{\left(n \cdot n\right)} \cdot {\ell}^{2}}{Om} \cdot \frac{U* \cdot U}{Om}\right)} \]

       6. unpow2 32.6%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\left(n \cdot n\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U* \cdot U}{Om}\right)} \]

       7. swap-sqr 37.5%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}}{Om} \cdot \frac{U* \cdot U}{Om}\right)} \]

       8. *-commutative 37.5%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{\color{blue}{U \cdot U*}}{Om}\right)} \]

   11. Simplified 37.5%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}} \]

   if -3.29999999999999997e105 < l < -4.30000000000000024e-235

   1. Initial program 58.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

   4. Simplified 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   if -4.30000000000000024e-235 < l < 3.69999999999999999e92

   1. Initial program 62.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 71.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 71.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate-+l- 71.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

      4. sub-neg 71.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      5. associate-/l* 71.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

      6. remove-double-neg 71.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

      7. associate-*l* 71.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 64.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 64.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   6. Simplified 64.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -4.3 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \end{array} \]

Alternative 13: 44.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}{U}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om)))
   (if (<= l -2.55e+107)
     (sqrt (* 2.0 (* (/ (* (* n l) (* n l)) Om) (/ (* U U*) Om))))
     (if (<= l -6e-225)
       (sqrt (* (* (* 2.0 n) U) (+ t (* t_1 -2.0))))
       (if (<= l 3.35e+103)
         (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1)))))
         (sqrt (* (* 2.0 n) (/ n (/ (/ (* Om Om) (* (* l l) U*)) U)))))))))

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 <= -2.55e+107) {
		tmp = sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	} else if (l <= -6e-225) {
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 3.35e+103) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))));
	}
	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 <= (-2.55d+107)) then
        tmp = sqrt((2.0d0 * ((((n * l) * (n * l)) / om) * ((u * u_42) / om))))
    else if (l <= (-6d-225)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (t_1 * (-2.0d0)))))
    else if (l <= 3.35d+103) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else
        tmp = sqrt(((2.0d0 * n) * (n / (((om * om) / ((l * l) * u_42)) / u))))
    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 <= -2.55e+107) {
		tmp = Math.sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	} else if (l <= -6e-225) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else if (l <= 3.35e+103) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	tmp = 0
	if l <= -2.55e+107:
		tmp = math.sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))))
	elif l <= -6e-225:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))))
	elif l <= 3.35e+103:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	tmp = 0.0
	if (l <= -2.55e+107)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(n * l) * Float64(n * l)) / Om) * Float64(Float64(U * U_42_) / Om))));
	elseif (l <= -6e-225)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(t_1 * -2.0))));
	elseif (l <= 3.35e+103)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(n / Float64(Float64(Float64(Om * Om) / Float64(Float64(l * l) * U_42_)) / U))));
	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 <= -2.55e+107)
		tmp = sqrt((2.0 * ((((n * l) * (n * l)) / Om) * ((U * U_42_) / Om))));
	elseif (l <= -6e-225)
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	elseif (l <= 3.35e+103)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	else
		tmp = sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, -2.55e+107], N[Sqrt[N[(2.0 * N[(N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -6e-225], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.35e+103], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(n / N[(N[(N[(Om * Om), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5500000000000001e107

   1. Initial program 14.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 17.5%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 17.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 17.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 17.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 17.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 30.4%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 30.4%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 30.4%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 30.4%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 30.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 30.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 30.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 52.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U around 0 39.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. div-inv 39.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot \frac{1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

   6. Applied egg-rr 39.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot \frac{1}{Om}}\right)}{Om}\right) \cdot U\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 39.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{\left(n \cdot \left(\ell \cdot U*\right)\right) \cdot 1}{Om}}\right)}{Om}\right) \cdot U\right)} \]

      2. *-rgt-identity 39.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{\color{blue}{n \cdot \left(\ell \cdot U*\right)}}{Om}\right)}{Om}\right) \cdot U\right)} \]

      3. associate-/l* 39.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{\frac{Om}{\ell \cdot U*}}}\right)}{Om}\right) \cdot U\right)} \]

      4. associate-/r/ 39.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   8. Simplified 39.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \color{blue}{\frac{n}{Om} \cdot \left(\ell \cdot U*\right)}\right)}{Om}\right) \cdot U\right)} \]

   9. Taylor expanded in n around inf 33.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{{n}^{2} \cdot \left({\ell}^{2} \cdot \left(U \cdot U*\right)\right)}{{Om}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-*r* 32.9%

          \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(U \cdot U*\right)}}{{Om}^{2}}} \]

       2. *-commutative 32.9%

          \[\leadsto \sqrt{2 \cdot \frac{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \color{blue}{\left(U* \cdot U\right)}}{{Om}^{2}}} \]

       3. unpow2 32.9%

          \[\leadsto \sqrt{2 \cdot \frac{\left({n}^{2} \cdot {\ell}^{2}\right) \cdot \left(U* \cdot U\right)}{\color{blue}{Om \cdot Om}}} \]

       4. times-frac 33.2%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\frac{{n}^{2} \cdot {\ell}^{2}}{Om} \cdot \frac{U* \cdot U}{Om}\right)}} \]

       5. unpow2 33.2%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\color{blue}{\left(n \cdot n\right)} \cdot {\ell}^{2}}{Om} \cdot \frac{U* \cdot U}{Om}\right)} \]

       6. unpow2 33.2%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\left(n \cdot n\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om} \cdot \frac{U* \cdot U}{Om}\right)} \]

       7. swap-sqr 41.9%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\color{blue}{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}}{Om} \cdot \frac{U* \cdot U}{Om}\right)} \]

       8. *-commutative 41.9%

          \[\leadsto \sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{\color{blue}{U \cdot U*}}{Om}\right)} \]

   11. Simplified 41.9%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}} \]

   if -2.5500000000000001e107 < l < -6.00000000000000035e-225

   1. Initial program 58.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

   4. Simplified 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   if -6.00000000000000035e-225 < l < 3.35000000000000017e103

   1. Initial program 63.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 72.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 72.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate-+l- 72.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

      4. sub-neg 72.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      5. associate-/l* 72.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

      6. remove-double-neg 72.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

      7. associate-*l* 72.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 65.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   6. Simplified 65.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right)} \]

   if 3.35000000000000017e103 < l

   1. Initial program 18.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 18.3%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 18.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 18.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 18.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 18.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 34.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 34.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 34.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 34.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 34.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 34.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 34.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U* around inf 26.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{n \cdot \left({\ell}^{2} \cdot \left(U* \cdot U\right)\right)}{{Om}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 26.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{n}{\frac{{Om}^{2}}{{\ell}^{2} \cdot \left(U* \cdot U\right)}}}} \]

      2. associate-*r* 29.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{{Om}^{2}}{\color{blue}{\left({\ell}^{2} \cdot U*\right) \cdot U}}}} \]

      3. associate-/r* 29.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\color{blue}{\frac{\frac{{Om}^{2}}{{\ell}^{2} \cdot U*}}{U}}}} \]

      4. unpow2 29.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{\color{blue}{Om \cdot Om}}{{\ell}^{2} \cdot U*}}{U}}} \]

      5. *-commutative 29.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{Om \cdot Om}{\color{blue}{U* \cdot {\ell}^{2}}}}{U}}} \]

      6. unpow2 29.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{Om \cdot Om}{U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}}}{U}}} \]

   6. Simplified 29.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\frac{n}{\frac{\frac{Om \cdot Om}{U* \cdot \left(\ell \cdot \ell\right)}}{U}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{U \cdot U*}{Om}\right)}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}{U}}}\\ \end{array} \]

Alternative 14: 42.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5.4e+40)
   (sqrt (* (* (* 2.0 n) U) t))
   (if (<= t 1.7e+82)
     (sqrt (* 2.0 (* U (* n (+ t (* (/ (* l l) Om) -2.0))))))
     (pow (* (* 2.0 n) (* U t)) 0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5.4e+40) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else if (t <= 1.7e+82) {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-5.4d+40)) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else if (t <= 1.7d+82) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) / om) * (-2.0d0)))))))
    else
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5.4e+40) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else if (t <= 1.7e+82) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5.4e+40:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	elif t <= 1.7e+82:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))))
	else:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5.4e+40)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	elseif (t <= 1.7e+82)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5.4e+40)
		tmp = sqrt((((2.0 * n) * U) * t));
	elseif (t <= 1.7e+82)
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	else
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5.4e+40], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.7e+82], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.40000000000000019e40

   1. Initial program 53.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf 50.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -5.40000000000000019e40 < t < 1.69999999999999997e82

   1. Initial program 46.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 49.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 49.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 49.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 49.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 49.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 54.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 49.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 49.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 51.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in U around 0 54.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right) \cdot U\right)}} \]

   5. Taylor expanded in n around 0 41.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 42.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)}} \]

      2. unpow2 42.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right) \cdot U\right)} \]

   7. Simplified 42.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) \cdot U\right)}} \]

   if 1.69999999999999997e82 < t

   1. Initial program 48.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 51.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 51.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 51.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 51.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 51.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 58.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 58.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 58.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 58.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 48.1%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 48.1%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 48.1%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 51.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 55.0%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

   6. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 15: 35.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{n \cdot -2}{\frac{Om}{U \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5.3e-269)
   (sqrt (* (* (* 2.0 n) U) t))
   (if (<= t 4.6e-116)
     (sqrt (/ (* n -2.0) (/ Om (* U (* 2.0 (* l l))))))
     (pow (* (* 2.0 n) (* U t)) 0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5.3e-269) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else if (t <= 4.6e-116) {
		tmp = sqrt(((n * -2.0) / (Om / (U * (2.0 * (l * l))))));
	} else {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-5.3d-269)) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else if (t <= 4.6d-116) then
        tmp = sqrt(((n * (-2.0d0)) / (om / (u * (2.0d0 * (l * l))))))
    else
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5.3e-269) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else if (t <= 4.6e-116) {
		tmp = Math.sqrt(((n * -2.0) / (Om / (U * (2.0 * (l * l))))));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5.3e-269:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	elif t <= 4.6e-116:
		tmp = math.sqrt(((n * -2.0) / (Om / (U * (2.0 * (l * l))))))
	else:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5.3e-269)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	elseif (t <= 4.6e-116)
		tmp = sqrt(Float64(Float64(n * -2.0) / Float64(Om / Float64(U * Float64(2.0 * Float64(l * l))))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5.3e-269)
		tmp = sqrt((((2.0 * n) * U) * t));
	elseif (t <= 4.6e-116)
		tmp = sqrt(((n * -2.0) / (Om / (U * (2.0 * (l * l))))));
	else
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5.3e-269], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 4.6e-116], N[Sqrt[N[(N[(n * -2.0), $MachinePrecision] / N[(Om / N[(U * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.2999999999999998e-269

   1. Initial program 53.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf 46.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -5.2999999999999998e-269 < t < 4.60000000000000003e-116

   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* 52.9%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 52.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 52.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 52.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 52.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 56.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 56.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 56.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 56.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 52.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 52.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 54.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around -inf 47.3%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)\right)}{Om}}} \]
   5. Step-by-step derivation

      1. associate-/l* 45.5%

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)}}}} \]

      2. associate-*r/ 45.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-2 \cdot n}{\frac{Om}{{\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)}}}} \]

      3. associate-*r* 45.5%

         \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\color{blue}{\left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right) \cdot U}}}} \]

      4. unpow2 45.5%

         \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right) \cdot U}}} \]

      5. mul-1-neg 45.5%

         \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\left(\left(\ell \cdot \ell\right) \cdot \left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right) \cdot U}}} \]

      6. unsub-neg 45.5%

         \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right) \cdot U}}} \]

      7. associate-/l* 41.7%

         \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\left(\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)\right) \cdot U}}} \]

   6. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\frac{-2 \cdot n}{\frac{Om}{\left(\left(\ell \cdot \ell\right) \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)\right) \cdot U}}}} \]

   7. Taylor expanded in n around 0 32.0%

      \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot U}}} \]
   8. Step-by-step derivation

      1. unpow2 32.0%

         \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U}}} \]

   9. Simplified 32.0%

      \[\leadsto \sqrt{\frac{-2 \cdot n}{\frac{Om}{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot U}}} \]

   if 4.60000000000000003e-116 < t

   1. Initial program 44.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 47.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 47.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 47.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 47.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 47.8%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 53.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 53.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 53.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 53.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 46.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 46.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 48.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 42.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 44.5%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

   6. Applied egg-rr 44.5%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{n \cdot -2}{\frac{Om}{U \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 16: 43.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + t_1 \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om)))
   (if (<= l -4.2e-227)
     (sqrt (* (* (* 2.0 n) U) (+ t (* t_1 -2.0))))
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1))))))))

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 <= -4.2e-227) {
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * l) / om
    if (l <= (-4.2d-227)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (t_1 * (-2.0d0)))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double tmp;
	if (l <= -4.2e-227) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	tmp = 0
	if l <= -4.2e-227:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	tmp = 0.0
	if (l <= -4.2e-227)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(t_1 * -2.0))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	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 <= -4.2e-227)
		tmp = sqrt((((2.0 * n) * U) * (t + (t_1 * -2.0))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, -4.2e-227], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.1999999999999999e-227

   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. Taylor expanded in Om around inf 38.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 38.9%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

   4. Simplified 38.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   if -4.1999999999999999e-227 < l

   1. Initial program 51.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate-+l- 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

      4. sub-neg 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

      5. associate-/l* 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

      6. remove-double-neg 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

      7. associate-*l* 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 53.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 53.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   6. Simplified 53.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 17: 43.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (* l l) Om)))))))

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))))));
}

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))))))
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))))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l* 49.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   3. associate-+l- 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)} \]

   4. sub-neg 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{\ell \cdot \ell}{Om} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\right)\right)} \]

   5. associate-/l* 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}} + \left(-\left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)} \]

   6. remove-double-neg 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)} \]

   7. associate-*l* 53.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

3. Simplified 53.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in Om around inf 44.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 44.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

6. Simplified 44.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right)} \]

7. Final simplification 44.2%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

Alternative 18: 34.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.4e-232)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* (* 2.0 n) (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.4e-232) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= (-1.4d-232)) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.4e-232) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -1.4e-232:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.4e-232)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -1.4e-232)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt(((2.0 * n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.4e-232], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.39999999999999996e-232

   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. Taylor expanded in t around inf 35.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -1.39999999999999996e-232 < l

   1. Initial program 51.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 58.1%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 62.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 53.0%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 53.0%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 54.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 59.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 45.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 19: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* (* 2.0 n) (* U t)) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow(((2.0 * n) * (U * t)), 0.5);
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = ((2.0d0 * n) * (u * t)) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow(((2.0 * n) * (U * t)), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow(((2.0 * n) * (U * t)), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = ((2.0 * n) * (U * t)) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 48.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l* 49.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+ 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/ 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l* 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l* 46.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow2 46.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l* 47.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified 54.2%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 37.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
5. Step-by-step derivation

   1. pow1/2 39.4%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

6. Applied egg-rr 39.4%

   \[\leadsto \color{blue}{{\left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right)}^{0.5}} \]

7. Final simplification 39.4%

   \[\leadsto {\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5} \]

Alternative 20: 34.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* 2.0 n) (* U t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((2.0 * n) * (U * t)));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((2.0d0 * n) * (u * t)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt(((2.0 * n) * (U * t)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((2.0 * n) * (U * t)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * n) * Float64(U * t)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((2.0 * n) * (U * t)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l* 49.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+ 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in 49.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/ 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l* 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative 53.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l* 46.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow2 46.4%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l* 47.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified 54.2%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 37.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

5. Final simplification 37.7%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)} \]

Reproduce

herbie shell --seed 2023171 
(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*))))))