Toniolo and Linder, Equation (13)

Percentage Accurate: 49.7% → 62.3%

Time: 24.3s

Alternatives: 18

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := n \cdot \left(U \cdot t\right)\\ t_3 := \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;n \leq 4 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{t_3}{\frac{Om}{t_1}} + t_2\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-208} \lor \neg \left(n \leq 2.9 \cdot 10^{-62}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t_2 + 2 \cdot \frac{t_1 \cdot \left(\ell \cdot -2 - \frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U l)))
        (t_2 (* n (* U t)))
        (t_3 (fma l -2.0 (* (* n (/ l Om)) (- U* U)))))
   (if (<= n 4e-287)
     (sqrt (* 2.0 (+ (/ t_3 (/ Om t_1)) t_2)))
     (if (or (<= n 8e-208) (not (<= n 2.9e-62)))
       (* (sqrt (* n 2.0)) (sqrt (* U (+ t (* (/ l Om) t_3)))))
       (sqrt
        (+
         (* 2.0 t_2)
         (*
          2.0
          (/ (* t_1 (- (* l -2.0) (/ (* n (* l (- U U*))) Om))) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double t_2 = n * (U * t);
	double t_3 = fma(l, -2.0, ((n * (l / Om)) * (U_42_ - U)));
	double tmp;
	if (n <= 4e-287) {
		tmp = sqrt((2.0 * ((t_3 / (Om / t_1)) + t_2)));
	} else if ((n <= 8e-208) || !(n <= 2.9e-62)) {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t + ((l / Om) * t_3))));
	} else {
		tmp = sqrt(((2.0 * t_2) + (2.0 * ((t_1 * ((l * -2.0) - ((n * (l * (U - U_42_))) / Om))) / Om))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	t_2 = Float64(n * Float64(U * t))
	t_3 = fma(l, -2.0, Float64(Float64(n * Float64(l / Om)) * Float64(U_42_ - U)))
	tmp = 0.0
	if (n <= 4e-287)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t_3 / Float64(Om / t_1)) + t_2)));
	elseif ((n <= 8e-208) || !(n <= 2.9e-62))
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * t_3)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * t_2) + Float64(2.0 * Float64(Float64(t_1 * Float64(Float64(l * -2.0) - Float64(Float64(n * Float64(l * Float64(U - U_42_))) / Om))) / Om))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * -2.0 + N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 4e-287], N[Sqrt[N[(2.0 * N[(N[(t$95$3 / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[n, 8e-208], N[Not[LessEqual[n, 2.9e-62]], $MachinePrecision]], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * t$95$2), $MachinePrecision] + N[(2.0 * N[(N[(t$95$1 * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n * N[(l * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < 4.00000000000000009e-287

   1. Initial program 44.5%

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

      1. associate-*l* 44.7%

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

      2. sub-neg 44.7%

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

      3. associate--l+ 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-rgt-neg-in 44.7%

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

      6. associate-*l/ 51.3%

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

      7. associate-*l* 51.3%

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

      8. *-commutative 51.3%

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

      9. *-commutative 51.3%

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

      10. associate-*l* 44.2%

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

      11. unpow2 44.2%

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

      12. associate-*l* 45.1%

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

   3. Simplified 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 t around inf 50.9%

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

      1. distribute-lft-out 50.9%

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

      2. *-commutative 50.9%

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

      3. associate-/l* 55.0%

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

      4. +-commutative 55.0%

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

      5. *-commutative 55.0%

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

      6. associate-*r* 57.2%

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

      7. *-commutative 57.2%

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

      8. associate-*r* 51.0%

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

      9. associate-*l/ 53.6%

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

      10. fma-udef 53.6%

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

      11. associate-*r* 60.6%

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

   6. Simplified 60.6%

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

   if 4.00000000000000009e-287 < n < 8.0000000000000008e-208 or 2.89999999999999986e-62 < n

   1. Initial program 57.5%

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

      1. associate-*l* 55.7%

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

      2. sub-neg 55.7%

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

      3. associate--l+ 55.7%

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

      4. *-commutative 55.7%

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

      5. distribute-rgt-neg-in 55.7%

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

      6. associate-*l/ 62.5%

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

      7. associate-*l* 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. associate-*l* 58.8%

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

      11. unpow2 58.8%

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

      12. associate-*l* 58.9%

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

   3. Simplified 61.4%

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

      1. sqrt-prod 71.2%

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

      2. *-commutative 71.2%

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

   5. Applied egg-rr 71.2%

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

      1. *-commutative 71.2%

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

      2. *-commutative 71.2%

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

      3. associate-*r* 78.1%

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

   7. Simplified 78.1%

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

   if 8.0000000000000008e-208 < n < 2.89999999999999986e-62

   1. Initial program 42.5%

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

      1. associate-*l* 51.1%

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

      2. sub-neg 51.1%

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

      3. associate--l+ 51.1%

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

      4. *-commutative 51.1%

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

      5. distribute-rgt-neg-in 51.1%

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

      6. associate-*l/ 51.1%

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

      7. associate-*l* 51.1%

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

      8. *-commutative 51.1%

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

      9. *-commutative 51.1%

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

      10. associate-*l* 51.1%

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

      11. unpow2 51.1%

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

      12. associate-*l* 55.5%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in t around inf 79.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-208} \lor \neg \left(n \leq 2.9 \cdot 10^{-62}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot -2 - \frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\\ \end{array} \]

Alternative 2: 62.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 - \frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* U (* n 2.0))
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (- U* U) (* n (pow (/ l Om) 2.0))))))))
   (if (<= t_1 0.0)
     (*
      (sqrt (* n 2.0))
      (sqrt (* U (+ t (* (/ l Om) (- (* l -2.0) (* (/ n Om) (* U l))))))))
     (if (<= t_1 2e+152)
       t_1
       (if (<= t_1 INFINITY)
         (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
         (sqrt
          (*
           (* n 2.0)
           (/ (fma l -2.0 (* (* n (/ l Om)) (- U* U))) (/ Om (* U l))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * pow((l / Om), 2.0))))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t + ((l / Om) * ((l * -2.0) - ((n / Om) * (U * l)))))));
	} else if (t_1 <= 2e+152) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else {
		tmp = sqrt(((n * 2.0) * (fma(l, -2.0, ((n * (l / Om)) * (U_42_ - U))) / (Om / (U * l)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(U_42_ - U) * Float64(n * (Float64(l / Om) ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) - Float64(Float64(n / Om) * Float64(U * l))))))));
	elseif (t_1 <= 2e+152)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(fma(l, -2.0, Float64(Float64(n * Float64(l / Om)) * Float64(U_42_ - U))) / Float64(Om / Float64(U * l)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+152], t$95$1, If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l * -2.0 + N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 7.5%

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

      1. associate-*l* 37.9%

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

      2. sub-neg 37.9%

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

      3. associate--l+ 37.9%

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

      4. *-commutative 37.9%

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

      5. distribute-rgt-neg-in 37.9%

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

      6. associate-*l/ 37.9%

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

      7. associate-*l* 37.9%

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

      8. *-commutative 37.9%

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

      9. *-commutative 37.9%

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

      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* 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in U* around 0 34.7%

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

      1. *-commutative 34.7%

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

      2. +-commutative 34.7%

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

      3. associate-/l* 34.7%

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

      4. +-commutative 34.7%

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

      5. mul-1-neg 34.7%

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

      6. unsub-neg 34.7%

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

      7. *-commutative 34.7%

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

      8. associate-/l* 34.7%

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

   6. Simplified 34.7%

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

      1. sqrt-prod 38.9%

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

      2. associate-/r/ 39.0%

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

      3. associate-/r/ 39.0%

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

      4. *-commutative 39.0%

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

   8. Applied egg-rr 39.0%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 97.2%

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

   if 2.0000000000000001e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 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. *-commutative 22.7%

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

      2. associate-*l* 24.0%

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

      3. associate-*l* 25.2%

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

      4. associate-*l* 25.2%

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

      5. sub-neg 25.2%

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

      6. +-commutative 25.2%

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

      7. *-commutative 25.2%

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

      8. distribute-rgt-neg-in 25.2%

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

      9. fma-def 25.2%

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

      10. associate-*r/ 38.9%

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

      11. metadata-eval 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in n around 0 23.6%

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

      1. associate-*r* 22.4%

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

      2. +-commutative 22.4%

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

      3. unpow2 22.4%

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

      4. associate-*r/ 38.7%

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

      5. *-commutative 38.7%

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

      6. fma-udef 38.7%

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

   6. Simplified 38.7%

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

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

   1. Initial program 0.0%

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

      1. associate-*l* 3.0%

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

      2. sub-neg 3.0%

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

      3. associate--l+ 3.0%

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

      4. *-commutative 3.0%

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

      5. distribute-rgt-neg-in 3.0%

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

      6. associate-*l/ 14.6%

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

      7. associate-*l* 14.6%

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

      8. *-commutative 14.6%

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

      9. *-commutative 14.6%

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

      10. associate-*l* 14.7%

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

      11. unpow2 14.7%

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

      12. associate-*l* 17.7%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in t around 0 43.7%

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

      1. associate-/l* 43.6%

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

      2. +-commutative 43.6%

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

      3. *-commutative 43.6%

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

      4. associate-*r* 53.9%

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

      5. *-commutative 53.9%

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

      6. associate-*r* 53.4%

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

      7. associate-*l/ 55.2%

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

      8. fma-udef 55.2%

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

      9. associate-*r* 57.4%

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

   6. Simplified 57.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 - \frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)}{\frac{Om}{U \cdot \ell}}}\\ \end{array} \]

Alternative 3: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.9e+152)
   (sqrt
    (*
     2.0
     (+
      (/ (fma l -2.0 (* (* n (/ l Om)) (- U* U))) (/ Om (* n (* U l))))
      (* n (* U t)))))
   (*
    (* l (sqrt 2.0))
    (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.9e+152) {
		tmp = sqrt((2.0 * ((fma(l, -2.0, ((n * (l / Om)) * (U_42_ - U))) / (Om / (n * (U * l)))) + (n * (U * t)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.9e+152)
		tmp = sqrt(Float64(2.0 * Float64(Float64(fma(l, -2.0, Float64(Float64(n * Float64(l / Om)) * Float64(U_42_ - U))) / Float64(Om / Float64(n * Float64(U * l)))) + Float64(n * Float64(U * t)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U)))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.9e+152], N[Sqrt[N[(2.0 * N[(N[(N[(l * -2.0 + N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.90000000000000011e152

   1. Initial program 52.6%

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

      1. associate-*l* 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/ 57.2%

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

      7. associate-*l* 57.2%

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

      8. *-commutative 57.2%

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

      9. *-commutative 57.2%

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

      10. associate-*l* 51.2%

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

      11. unpow2 51.2%

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

      12. associate-*l* 52.3%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in t around inf 57.0%

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

      1. distribute-lft-out 57.0%

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

      2. *-commutative 57.0%

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

      3. associate-/l* 57.9%

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

      4. +-commutative 57.9%

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

      5. *-commutative 57.9%

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

      6. associate-*r* 59.3%

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

      7. *-commutative 59.3%

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

      8. associate-*r* 54.8%

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

      9. associate-*l/ 56.9%

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

      10. fma-udef 56.9%

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

      11. associate-*r* 62.8%

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

   6. Simplified 62.8%

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

   if 3.90000000000000011e152 < l

   1. Initial program 18.8%

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

      1. associate-*l* 19.2%

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

      2. sub-neg 19.2%

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

      3. associate--l+ 19.2%

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

      4. *-commutative 19.2%

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

      5. distribute-rgt-neg-in 19.2%

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

      6. associate-*l/ 38.9%

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

      7. associate-*l* 38.9%

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

      8. *-commutative 38.9%

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

      9. *-commutative 38.9%

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

      10. associate-*l* 38.9%

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

      11. unpow2 38.9%

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

      12. associate-*l* 39.0%

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

   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 t around inf 38.4%

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

      1. distribute-lft-out 38.4%

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

      2. *-commutative 38.4%

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

      3. associate-/l* 47.9%

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

      4. +-commutative 47.9%

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

      5. *-commutative 47.9%

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

      6. associate-*r* 52.1%

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

      7. *-commutative 52.1%

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

      8. associate-*r* 52.1%

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

      9. associate-*l/ 51.8%

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

      10. fma-udef 51.8%

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

      11. associate-*r* 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in l around inf 73.9%

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

      1. associate-/l* 77.2%

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

      2. *-commutative 77.2%

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

      3. sub-neg 77.2%

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

      4. associate-/l* 77.2%

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

      5. metadata-eval 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(\ell, -2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}} + n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\\ \end{array} \]

Alternative 4: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+143}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.8e+143)
   (pow
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (/ Om (* n (* U l))))))
    0.5)
   (*
    (* l (sqrt 2.0))
    (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.8e+143) {
		tmp = pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 + (n / (Om / (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) :: tmp
    if (l <= 2.8d+143) then
        tmp = (2.0d0 * ((n * (u * t)) + (((l * (-2.0d0)) + (n / ((om / l) / u_42))) / (om / (n * (u * l)))))) ** 0.5d0
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n / (om / (u * ((-2.0d0) + (n / (om / (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 tmp;
	if (l <= 2.8e+143) {
		tmp = Math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.8e+143:
		tmp = math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5)
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.8e+143)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.8e+143)
		tmp = (2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))) ^ 0.5;
	else
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.8e+143], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.79999999999999998e143

   1. Initial program 52.9%

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

      1. associate-*l* 53.2%

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

      2. sub-neg 53.2%

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

      3. associate--l+ 53.2%

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

      4. *-commutative 53.2%

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

      5. distribute-rgt-neg-in 53.2%

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

      6. associate-*l/ 57.5%

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

      7. associate-*l* 57.5%

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

      8. *-commutative 57.5%

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

      9. *-commutative 57.5%

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

      10. associate-*l* 51.4%

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

      11. unpow2 51.4%

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

      12. associate-*l* 52.5%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in t around inf 57.3%

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

      1. pow1/2 57.4%

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

      2. distribute-lft-out 57.4%

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

      3. associate-/l* 58.4%

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

      4. associate-/l* 57.5%

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

      5. *-commutative 57.5%

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

      6. *-commutative 57.5%

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

   6. Applied egg-rr 57.5%

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

   7. Taylor expanded in U* around inf 57.2%

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

      1. associate-/r* 61.1%

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

   9. Simplified 61.1%

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

   if 2.79999999999999998e143 < l

   1. Initial program 20.3%

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

      1. associate-*l* 20.7%

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

      2. sub-neg 20.7%

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

      3. associate--l+ 20.7%

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

      4. *-commutative 20.7%

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

      5. distribute-rgt-neg-in 20.7%

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

      6. associate-*l/ 38.5%

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

      7. associate-*l* 38.5%

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

      8. *-commutative 38.5%

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

      9. *-commutative 38.5%

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

      10. associate-*l* 38.5%

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

      11. unpow2 38.5%

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

      12. associate-*l* 38.6%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in t around inf 38.0%

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

      1. distribute-lft-out 38.0%

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

      2. *-commutative 38.0%

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

      3. associate-/l* 46.6%

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

      4. +-commutative 46.6%

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

      5. *-commutative 46.6%

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

      6. associate-*r* 53.6%

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

      7. *-commutative 53.6%

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

      8. associate-*r* 53.6%

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

      9. associate-*l/ 53.2%

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

      10. fma-udef 53.2%

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

      11. associate-*r* 53.2%

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

   6. Simplified 53.2%

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

   7. Taylor expanded in l around inf 67.6%

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

      1. associate-/l* 73.4%

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

      2. *-commutative 73.4%

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

      3. sub-neg 73.4%

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

      4. associate-/l* 73.5%

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

      5. metadata-eval 73.5%

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

   9. Simplified 73.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+143}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\\ \end{array} \]

Alternative 5: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{+137}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.6e+137)
   (pow
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (/ Om (* n (* U l))))))
    0.5)
   (*
    (* l (sqrt 2.0))
    (sqrt (/ n (/ Om (* U (- -2.0 (/ (* n (- U U*)) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.6e+137) {
		tmp = pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 - ((n * (U - U_42_)) / 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 <= 8.6d+137) then
        tmp = (2.0d0 * ((n * (u * t)) + (((l * (-2.0d0)) + (n / ((om / l) / u_42))) / (om / (n * (u * l)))))) ** 0.5d0
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n / (om / (u * ((-2.0d0) - ((n * (u - u_42)) / 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 <= 8.6e+137) {
		tmp = Math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n / (Om / (U * (-2.0 - ((n * (U - U_42_)) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.6e+137:
		tmp = math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5)
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n / (Om / (U * (-2.0 - ((n * (U - U_42_)) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.6e+137)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 - Float64(Float64(n * Float64(U - U_42_)) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8.6e+137)
		tmp = (2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))) ^ 0.5;
	else
		tmp = (l * sqrt(2.0)) * sqrt((n / (Om / (U * (-2.0 - ((n * (U - U_42_)) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.6e+137], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 - N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.59999999999999929e137

   1. Initial program 52.7%

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

      1. associate-*l* 53.0%

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

      2. sub-neg 53.0%

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

      3. associate--l+ 53.0%

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

      4. *-commutative 53.0%

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

      5. distribute-rgt-neg-in 53.0%

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

      6. associate-*l/ 57.4%

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

      7. associate-*l* 57.4%

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

      8. *-commutative 57.4%

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

      9. *-commutative 57.4%

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

      10. associate-*l* 51.2%

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

      11. unpow2 51.2%

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

      12. associate-*l* 52.3%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in t around inf 57.6%

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

      1. pow1/2 57.8%

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

      2. distribute-lft-out 57.8%

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

      3. associate-/l* 58.7%

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

      4. associate-/l* 57.9%

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

      5. *-commutative 57.9%

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

      6. *-commutative 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in U* around inf 57.5%

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

      1. associate-/r* 61.0%

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

   9. Simplified 61.0%

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

   if 8.59999999999999929e137 < l

   1. Initial program 24.2%

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

      1. associate-*l* 24.5%

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

      2. sub-neg 24.5%

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

      3. associate--l+ 24.5%

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

      4. *-commutative 24.5%

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

      5. distribute-rgt-neg-in 24.5%

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

      6. associate-*l/ 40.9%

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

      7. associate-*l* 40.9%

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

      8. *-commutative 40.9%

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

      9. *-commutative 40.9%

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

      10. associate-*l* 40.9%

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

      11. unpow2 40.9%

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

      12. associate-*l* 41.0%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in t around inf 37.6%

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

      1. distribute-lft-out 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/l* 45.5%

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

      4. +-commutative 45.5%

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

      5. *-commutative 45.5%

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

      6. associate-*r* 54.7%

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

      7. *-commutative 54.7%

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

      8. associate-*r* 54.7%

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

      9. associate-*l/ 54.4%

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

      10. fma-udef 54.4%

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

      11. associate-*r* 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in l around inf 64.9%

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

      1. associate-/l* 73.0%

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

      2. *-commutative 73.0%

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

      3. sub-neg 73.0%

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

      4. metadata-eval 73.0%

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

   9. Simplified 73.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{+137}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)}}}\\ \end{array} \]

Alternative 6: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.35e+154)
   (pow
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (/ Om (* n (* U l))))))
    0.5)
   (* (* l (sqrt 2.0)) (sqrt (* (/ n Om) (* U (+ -2.0 (/ n (/ Om U*)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.35e+154) {
		tmp = pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + (n / (Om / U_42_))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.35d+154) then
        tmp = (2.0d0 * ((n * (u * t)) + (((l * (-2.0d0)) + (n / ((om / l) / u_42))) / (om / (n * (u * l)))))) ** 0.5d0
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((n / om) * (u * ((-2.0d0) + (n / (om / u_42))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.35e+154) {
		tmp = Math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((n / Om) * (U * (-2.0 + (n / (Om / U_42_))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.35e+154:
		tmp = math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5)
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((n / Om) * (U * (-2.0 + (n / (Om / U_42_))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.35e+154)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n / Om) * Float64(U * Float64(-2.0 + Float64(n / Float64(Om / U_42_)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.35e+154)
		tmp = (2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))) ^ 0.5;
	else
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + (n / (Om / U_42_))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.35e+154], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.34999999999999992e154

   1. Initial program 52.9%

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

      1. associate-*l* 52.7%

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

      2. sub-neg 52.7%

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

      3. associate--l+ 52.7%

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

      4. *-commutative 52.7%

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

      5. distribute-rgt-neg-in 52.7%

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

      6. associate-*l/ 57.0%

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

      7. associate-*l* 57.0%

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

      8. *-commutative 57.0%

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

      9. *-commutative 57.0%

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

      10. associate-*l* 51.0%

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

      11. unpow2 51.0%

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

      12. associate-*l* 52.0%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in t around inf 56.7%

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

      1. pow1/2 56.9%

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

      2. distribute-lft-out 56.9%

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

      3. associate-/l* 57.8%

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

      4. associate-/l* 57.0%

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

      5. *-commutative 57.0%

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

      6. *-commutative 57.0%

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

   6. Applied egg-rr 57.0%

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

   7. Taylor expanded in U* around inf 56.6%

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

      1. associate-/r* 60.5%

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

   9. Simplified 60.5%

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

   if 2.34999999999999992e154 < l

   1. Initial program 16.1%

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

      1. associate-*l* 19.7%

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

      2. sub-neg 19.7%

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

      3. associate--l+ 19.7%

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

      4. *-commutative 19.7%

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

      5. distribute-rgt-neg-in 19.7%

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

      6. associate-*l/ 40.0%

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

      7. associate-*l* 40.0%

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

      8. *-commutative 40.0%

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

      9. *-commutative 40.0%

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

      10. associate-*l* 40.0%

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

      11. unpow2 40.0%

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

      12. associate-*l* 40.2%

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

   3. Simplified 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 t around inf 39.7%

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

      1. distribute-lft-out 39.7%

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

      2. *-commutative 39.7%

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

      3. associate-/l* 49.5%

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

      4. +-commutative 49.5%

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

      5. *-commutative 49.5%

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

      6. associate-*r* 53.8%

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

      7. *-commutative 53.8%

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

      8. associate-*r* 53.8%

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

      9. associate-*l/ 53.4%

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

      10. fma-udef 53.4%

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

      11. associate-*r* 53.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in l around inf 76.3%

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

      1. associate-/l* 79.7%

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

      2. *-commutative 79.7%

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

      3. sub-neg 79.7%

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

      4. associate-/l* 79.7%

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

      5. metadata-eval 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in U around 0 76.1%

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

       1. *-un-lft-identity 76.1%

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

       2. associate-/l* 79.5%

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

       3. *-commutative 79.5%

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

       4. sub-neg 79.5%

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

       5. associate-/l* 79.5%

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

       6. metadata-eval 79.5%

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

   12. Applied egg-rr 79.5%

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

       1. *-lft-identity 79.5%

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

       2. associate-/r/ 72.7%

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

       3. +-commutative 72.7%

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

   14. Simplified 72.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n}{\frac{Om}{U*}}\right)\right)}\\ \end{array} \]

Alternative 7: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{+150}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.4e+150)
   (pow
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (/ Om (* n (* U l))))))
    0.5)
   (* (* 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 <= 6.4e+150) {
		tmp = pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} 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 <= 6.4d+150) then
        tmp = (2.0d0 * ((n * (u * t)) + (((l * (-2.0d0)) + (n / ((om / l) / u_42))) / (om / (n * (u * l)))))) ** 0.5d0
    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 <= 6.4e+150) {
		tmp = Math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
	} 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 <= 6.4e+150:
		tmp = math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5)
	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 <= 6.4e+150)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	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 <= 6.4e+150)
		tmp = (2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))) ^ 0.5;
	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, 6.4e+150], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $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 2 regimes

2. if l < 6.40000000000000031e150

   1. Initial program 52.6%

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

      1. associate-*l* 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/ 57.2%

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

      7. associate-*l* 57.2%

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

      8. *-commutative 57.2%

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

      9. *-commutative 57.2%

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

      10. associate-*l* 51.2%

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

      11. unpow2 51.2%

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

      12. associate-*l* 52.3%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in t around inf 57.0%

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

      1. pow1/2 57.1%

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

      2. distribute-lft-out 57.1%

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

      3. associate-/l* 58.1%

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

      4. associate-/l* 57.2%

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

      5. *-commutative 57.2%

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

      6. *-commutative 57.2%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in U* around inf 56.9%

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

      1. associate-/r* 60.7%

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

   9. Simplified 60.7%

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

   if 6.40000000000000031e150 < l

   1. Initial program 18.8%

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

      1. associate-*l* 19.2%

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

      2. sub-neg 19.2%

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

      3. associate--l+ 19.2%

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

      4. *-commutative 19.2%

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

      5. distribute-rgt-neg-in 19.2%

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

      6. associate-*l/ 38.9%

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

      7. associate-*l* 38.9%

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

      8. *-commutative 38.9%

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

      9. *-commutative 38.9%

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

      10. associate-*l* 38.9%

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

      11. unpow2 38.9%

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

      12. associate-*l* 39.0%

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

   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 t around inf 38.4%

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

      1. distribute-lft-out 38.4%

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

      2. *-commutative 38.4%

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

      3. associate-/l* 47.9%

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

      4. +-commutative 47.9%

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

      5. *-commutative 47.9%

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

      6. associate-*r* 52.1%

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

      7. *-commutative 52.1%

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

      8. associate-*r* 52.1%

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

      9. associate-*l/ 51.8%

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

      10. fma-udef 51.8%

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

      11. associate-*r* 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in l around inf 73.9%

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

      1. associate-/l* 77.2%

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

      2. *-commutative 77.2%

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

      3. sub-neg 77.2%

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

      4. associate-/l* 77.2%

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

      5. metadata-eval 77.2%

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

   9. Simplified 77.2%

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

   10. Taylor expanded in U around 0 73.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{+150}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 8: 58.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{-245} \lor \neg \left(n \leq 9 \cdot 10^{-178}\right):\\ \;\;\;\;{\left(2 \cdot \left(t_2 + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{t_1}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t_2 - \frac{t_1 \cdot \left(\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2\right)}{Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U l))) (t_2 (* n (* U t))))
   (if (or (<= n -2.2e-245) (not (<= n 9e-178)))
     (pow
      (* 2.0 (+ t_2 (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (/ Om t_1))))
      0.5)
     (sqrt
      (-
       (* 2.0 t_2)
       (* (/ (* t_1 (- (/ (* l (* n U)) Om) (* l -2.0))) Om) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double t_2 = n * (U * t);
	double tmp;
	if ((n <= -2.2e-245) || !(n <= 9e-178)) {
		tmp = pow((2.0 * (t_2 + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / t_1)))), 0.5);
	} else {
		tmp = sqrt(((2.0 * t_2) - (((t_1 * (((l * (n * U)) / Om) - (l * -2.0))) / Om) * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = n * (u * l)
    t_2 = n * (u * t)
    if ((n <= (-2.2d-245)) .or. (.not. (n <= 9d-178))) then
        tmp = (2.0d0 * (t_2 + (((l * (-2.0d0)) + (n / ((om / l) / u_42))) / (om / t_1)))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * t_2) - (((t_1 * (((l * (n * u)) / om) - (l * (-2.0d0)))) / om) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double t_2 = n * (U * t);
	double tmp;
	if ((n <= -2.2e-245) || !(n <= 9e-178)) {
		tmp = Math.pow((2.0 * (t_2 + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / t_1)))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * t_2) - (((t_1 * (((l * (n * U)) / Om) - (l * -2.0))) / Om) * 2.0)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U * l)
	t_2 = n * (U * t)
	tmp = 0
	if (n <= -2.2e-245) or not (n <= 9e-178):
		tmp = math.pow((2.0 * (t_2 + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / t_1)))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * t_2) - (((t_1 * (((l * (n * U)) / Om) - (l * -2.0))) / Om) * 2.0)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	t_2 = Float64(n * Float64(U * t))
	tmp = 0.0
	if ((n <= -2.2e-245) || !(n <= 9e-178))
		tmp = Float64(2.0 * Float64(t_2 + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om / t_1)))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * t_2) - Float64(Float64(Float64(t_1 * Float64(Float64(Float64(l * Float64(n * U)) / Om) - Float64(l * -2.0))) / Om) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U * l);
	t_2 = n * (U * t);
	tmp = 0.0;
	if ((n <= -2.2e-245) || ~((n <= 9e-178)))
		tmp = (2.0 * (t_2 + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / t_1)))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * t_2) - (((t_1 * (((l * (n * U)) / Om) - (l * -2.0))) / Om) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[n, -2.2e-245], N[Not[LessEqual[n, 9e-178]], $MachinePrecision]], N[Power[N[(2.0 * N[(t$95$2 + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * t$95$2), $MachinePrecision] - N[(N[(N[(t$95$1 * N[(N[(N[(l * N[(n * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.19999999999999993e-245 or 8.99999999999999957e-178 < n

   1. Initial program 52.7%

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

      1. associate-*l* 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/ 58.6%

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

      7. associate-*l* 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. associate-*l* 52.1%

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

      11. unpow2 52.1%

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

      12. associate-*l* 52.7%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in t around inf 56.8%

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

      1. pow1/2 57.0%

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

      2. distribute-lft-out 57.0%

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

      3. associate-/l* 60.4%

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

      4. associate-/l* 60.4%

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

      5. *-commutative 60.4%

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

      6. *-commutative 60.4%

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

   6. Applied egg-rr 60.4%

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

   7. Taylor expanded in U* around inf 60.3%

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

      1. associate-/r* 64.6%

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

   9. Simplified 64.6%

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

   if -2.19999999999999993e-245 < n < 8.99999999999999957e-178

   1. Initial program 31.7%

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

      1. associate-*l* 32.2%

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

      2. sub-neg 32.2%

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

      3. associate--l+ 32.2%

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

      4. *-commutative 32.2%

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

      5. distribute-rgt-neg-in 32.2%

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

      6. associate-*l/ 40.1%

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

      7. associate-*l* 40.1%

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

      8. *-commutative 40.1%

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

      9. *-commutative 40.1%

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

      10. associate-*l* 40.1%

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

      11. unpow2 40.1%

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

      12. associate-*l* 42.3%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in t around inf 46.3%

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

   5. Taylor expanded in U* around 0 54.3%

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

      1. associate-*r/ 54.3%

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

      2. mul-1-neg 54.3%

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

      3. *-commutative 54.3%

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

      4. associate-*r* 54.7%

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

      5. distribute-rgt-neg-in 54.7%

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

   7. Simplified 54.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{-245} \lor \neg \left(n \leq 9 \cdot 10^{-178}\right):\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{\ell \cdot \left(n \cdot U\right)}{Om} - \ell \cdot -2\right)}{Om} \cdot 2}\\ \end{array} \]

Alternative 9: 57.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\ell \cdot -2 + \frac{n}{\frac{\frac{Om}{\ell}}{U*}}}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (pow
  (*
   2.0
   (+
    (* n (* U t))
    (/ (+ (* l -2.0) (/ n (/ (/ Om l) U*))) (/ Om (* n (* U l))))))
  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)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 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)) + (((l * (-2.0d0)) + (n / ((om / l) / u_42))) / (om / (n * (u * l)))))) ** 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)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(l * -2.0) + Float64(n / Float64(Float64(Om / l) / U_42_))) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * ((n * (U * t)) + (((l * -2.0) + (n / ((Om / l) / U_42_))) / (Om / (n * (U * l)))))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(n / N[(N[(Om / l), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. associate-*l* 49.0%

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

   2. sub-neg 49.0%

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

   3. associate--l+ 49.0%

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

   4. *-commutative 49.0%

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

   5. distribute-rgt-neg-in 49.0%

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

   6. associate-*l/ 55.1%

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

   7. associate-*l* 55.1%

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

   8. *-commutative 55.1%

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

   9. *-commutative 55.1%

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

   10. associate-*l* 49.8%

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

   11. unpow2 49.8%

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

   12. associate-*l* 50.7%

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

3. Simplified 54.4%

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

4. Taylor expanded in t around inf 54.8%

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

   1. pow1/2 55.0%

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

   2. distribute-lft-out 55.0%

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

   3. associate-/l* 57.0%

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

   4. associate-/l* 56.2%

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

   5. *-commutative 56.2%

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

   6. *-commutative 56.2%

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in U* around inf 55.9%

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

   1. associate-/r* 59.4%

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

9. Simplified 59.4%

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

10. Final simplification 59.4%

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

Alternative 10: 57.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -8.5 \cdot 10^{+165} \lor \neg \left(Om \leq 1.05 \cdot 10^{+109}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -8.5e+165) (not (<= Om 1.05e+109)))
   (sqrt (* (* n 2.0) (* U (- t (* (/ l Om) (* 2.0 l))))))
   (sqrt
    (*
     (* n 2.0)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -8.5e+165) || !(Om <= 1.05e+109)) {
		tmp = sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-8.5d+165)) .or. (.not. (om <= 1.05d+109))) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - ((l / om) * (2.0d0 * l))))))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -8.5e+165) || !(Om <= 1.05e+109)) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -8.5e+165) or not (Om <= 1.05e+109):
		tmp = math.sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -8.5e+165) || !(Om <= 1.05e+109))
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -8.5e+165) || ~((Om <= 1.05e+109)))
		tmp = sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))));
	else
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -8.5e+165], N[Not[LessEqual[Om, 1.05e+109]], $MachinePrecision]], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -8.5000000000000001e165 or 1.0500000000000001e109 < Om

   1. Initial program 50.5%

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

      1. associate-*l* 48.0%

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

      2. sub-neg 48.0%

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

      3. associate-+l- 48.0%

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

      4. sub-neg 48.0%

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

      5. associate-/l* 61.3%

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

      6. remove-double-neg 61.3%

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

      7. associate-*l* 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in Om around inf 46.1%

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

      1. unpow2 46.1%

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

      2. associate-*r/ 58.5%

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

      3. associate-*r* 58.5%

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

   6. Simplified 58.5%

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

   if -8.5000000000000001e165 < Om < 1.0500000000000001e109

   1. Initial program 47.5%

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

      1. associate-*l* 49.6%

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

      2. sub-neg 49.6%

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

      3. associate--l+ 49.6%

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

      4. *-commutative 49.6%

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

      5. distribute-rgt-neg-in 49.6%

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

      6. associate-*l/ 50.9%

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

      7. associate-*l* 50.9%

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

      8. *-commutative 50.9%

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

      9. *-commutative 50.9%

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

      10. associate-*l* 47.0%

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

      11. unpow2 47.0%

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

      12. associate-*l* 47.9%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in U around 0 54.8%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8.5 \cdot 10^{+165} \lor \neg \left(Om \leq 1.05 \cdot 10^{+109}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 11: 51.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -4.6 \cdot 10^{-134} \lor \neg \left(Om \leq 8.8 \cdot 10^{-103}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -4.6e-134) (not (<= Om 8.8e-103)))
   (sqrt (* (* n 2.0) (* U (- t (* (/ l Om) (* 2.0 l))))))
   (sqrt
    (* 2.0 (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l U*)) Om))))) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -4.6e-134) || !(Om <= 8.8e-103)) {
		tmp = sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))));
	} else {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-4.6d-134)) .or. (.not. (om <= 8.8d-103))) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - ((l / om) * (2.0d0 * l))))))
    else
        tmp = sqrt((2.0d0 * ((n * (l * (u * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))))) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -4.6e-134) || !(Om <= 8.8e-103)) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))));
	} else {
		tmp = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -4.6e-134) or not (Om <= 8.8e-103):
		tmp = math.sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))))
	else:
		tmp = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -4.6e-134) || !(Om <= 8.8e-103))
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -4.6e-134) || ~((Om <= 8.8e-103)))
		tmp = sqrt(((n * 2.0) * (U * (t - ((l / Om) * (2.0 * l))))));
	else
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -4.6e-134], N[Not[LessEqual[Om, 8.8e-103]], $MachinePrecision]], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.6000000000000001e-134 or 8.7999999999999997e-103 < Om

   1. Initial program 51.4%

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

      1. associate-*l* 51.6%

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

      2. sub-neg 51.6%

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

      3. associate-+l- 51.6%

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

      4. sub-neg 51.6%

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

      5. associate-/l* 59.0%

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

      6. remove-double-neg 59.0%

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

      7. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in Om around inf 49.2%

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

      1. unpow2 49.2%

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

      2. associate-*r/ 55.2%

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

      3. associate-*r* 55.2%

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

   6. Simplified 55.2%

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

   if -4.6000000000000001e-134 < Om < 8.7999999999999997e-103

   1. Initial program 36.4%

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

      1. associate-*l* 36.8%

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

      2. sub-neg 36.8%

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

      3. associate--l+ 36.8%

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

      4. *-commutative 36.8%

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

      5. distribute-rgt-neg-in 36.8%

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

      6. associate-*l/ 36.8%

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

      7. associate-*l* 36.8%

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

      8. *-commutative 36.8%

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

      9. *-commutative 36.8%

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

      10. associate-*l* 34.7%

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

      11. unpow2 34.7%

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

      12. associate-*l* 34.9%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in t around 0 50.2%

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

   5. Taylor expanded in U around 0 50.6%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.6 \cdot 10^{-134} \lor \neg \left(Om \leq 8.8 \cdot 10^{-103}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 12: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.6e+116)
   (pow (* n (* 2.0 (* U t))) 0.5)
   (sqrt (* 2.0 (/ (* -2.0 (* n (* U (* l l)))) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.6e+116) {
		tmp = pow((n * (2.0 * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * ((-2.0 * (n * (U * (l * l)))) / Om)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.6d+116) then
        tmp = (n * (2.0d0 * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (((-2.0d0) * (n * (u * (l * l)))) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.6e+116) {
		tmp = Math.pow((n * (2.0 * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * ((-2.0 * (n * (U * (l * l)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.6e+116:
		tmp = math.pow((n * (2.0 * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * ((-2.0 * (n * (U * (l * l)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.6e+116)
		tmp = Float64(n * Float64(2.0 * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(-2.0 * Float64(n * Float64(U * Float64(l * l)))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.6e+116)
		tmp = (n * (2.0 * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * ((-2.0 * (n * (U * (l * l)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.6e+116], N[Power[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(-2.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.59999999999999987e116

   1. Initial program 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*l* 53.1%

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

      3. associate-*l* 53.0%

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

      4. associate-*l* 51.7%

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

      5. sub-neg 51.7%

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

      6. +-commutative 51.7%

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

      7. *-commutative 51.7%

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

      8. distribute-rgt-neg-in 51.7%

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

      9. fma-def 51.7%

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

      10. associate-*r/ 56.2%

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

      11. metadata-eval 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in l around 0 45.0%

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

      1. pow1/2 46.0%

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

      2. associate-*l* 46.0%

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

   6. Applied egg-rr 46.0%

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

   if 2.59999999999999987e116 < l

   1. Initial program 27.0%

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

      1. associate-*l* 29.7%

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

      2. sub-neg 29.7%

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

      3. associate--l+ 29.7%

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

      4. *-commutative 29.7%

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

      5. distribute-rgt-neg-in 29.7%

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

      6. associate-*l/ 44.4%

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

      7. associate-*l* 44.4%

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

      8. *-commutative 44.4%

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

      9. *-commutative 44.4%

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

      10. associate-*l* 44.4%

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

      11. unpow2 44.4%

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

      12. associate-*l* 44.5%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in t around 0 39.5%

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

   5. Taylor expanded in n around 0 25.3%

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

      1. *-commutative 25.3%

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

      2. unpow2 25.3%

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

   7. Simplified 25.3%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{-2 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 13: 43.4% accurate, 2.0× speedup

Math

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

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(2.0 * Float64(Float64(n * 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[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. associate-*l* 49.0%

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

   2. sub-neg 49.0%

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

   3. associate--l+ 49.0%

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

   4. *-commutative 49.0%

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

   5. distribute-rgt-neg-in 49.0%

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

   6. associate-*l/ 55.1%

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

   7. associate-*l* 55.1%

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

   8. *-commutative 55.1%

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

   9. *-commutative 55.1%

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

   10. associate-*l* 49.8%

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

   11. unpow2 49.8%

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

   12. associate-*l* 50.7%

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

3. Simplified 54.4%

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

4. Taylor expanded in U* around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. +-commutative 40.7%

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

   3. associate-/l* 45.4%

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

   4. +-commutative 45.4%

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

   5. mul-1-neg 45.4%

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

   6. unsub-neg 45.4%

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

   7. *-commutative 45.4%

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

   8. associate-/l* 45.9%

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

6. Simplified 45.9%

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

7. Taylor expanded in U around 0 45.0%

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

   1. *-commutative 45.0%

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

   2. unpow2 45.0%

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

9. Simplified 45.0%

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

10. Final simplification 45.0%

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

Alternative 14: 46.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. associate-*l* 49.0%

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

   2. sub-neg 49.0%

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

   3. associate-+l- 49.0%

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

   4. sub-neg 49.0%

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

   5. associate-/l* 55.0%

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

   6. remove-double-neg 55.0%

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

   7. associate-*l* 53.9%

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

3. Simplified 53.9%

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

4. Taylor expanded in Om around inf 45.3%

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

   1. unpow2 45.3%

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

   2. associate-*r/ 50.3%

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

   3. associate-*r* 50.3%

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

6. Simplified 50.3%

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

7. Final simplification 50.3%

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

Alternative 15: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.3e-101)
   (sqrt (* n (* t (* 2.0 U))))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.3e-101) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.3d-101) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.3e-101) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.3e-101:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.3e-101)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.3e-101)
		tmp = sqrt((n * (t * (2.0 * U))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.3e-101], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2999999999999997e-101

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 50.0%

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

      3. associate-*l* 50.9%

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

      4. associate-*l* 50.4%

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

      5. sub-neg 50.4%

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

      6. +-commutative 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-rgt-neg-in 50.4%

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

      9. fma-def 50.4%

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

      10. associate-*r/ 55.9%

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

      11. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in l around 0 43.9%

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

   if 4.2999999999999997e-101 < l

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-*l* 46.0%

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

      4. associate-*l* 43.6%

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

      5. sub-neg 43.6%

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

      6. +-commutative 43.6%

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

      7. *-commutative 43.6%

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

      8. distribute-rgt-neg-in 43.6%

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

      9. fma-def 43.6%

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

      10. associate-*r/ 50.8%

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

      11. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in l around 0 29.0%

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

      1. pow1/2 30.3%

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

      2. associate-*l* 30.3%

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

   6. Applied egg-rr 30.3%

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

   7. Taylor expanded in n around 0 30.3%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]
   8. Step-by-step derivation

      1. associate-*r* 36.3%

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

   9. Simplified 36.3%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot t\right) \cdot U\right)\right)}}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 16: 37.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-101}:\\ \;\;\;\;{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4e-101)
   (pow (* n (* 2.0 (* U t))) 0.5)
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e-101) {
		tmp = pow((n * (2.0 * (U * t))), 0.5);
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4d-101) then
        tmp = (n * (2.0d0 * (u * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4e-101) {
		tmp = Math.pow((n * (2.0 * (U * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4e-101:
		tmp = math.pow((n * (2.0 * (U * t))), 0.5)
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4e-101)
		tmp = Float64(n * Float64(2.0 * Float64(U * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4e-101)
		tmp = (n * (2.0 * (U * t))) ^ 0.5;
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4e-101], N[Power[N[(n * N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000021e-101

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 50.0%

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

      3. associate-*l* 50.9%

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

      4. associate-*l* 50.4%

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

      5. sub-neg 50.4%

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

      6. +-commutative 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-rgt-neg-in 50.4%

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

      9. fma-def 50.4%

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

      10. associate-*r/ 55.9%

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

      11. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in l around 0 43.9%

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

      1. pow1/2 45.1%

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

      2. associate-*l* 45.1%

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

   6. Applied egg-rr 45.1%

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

   if 4.00000000000000021e-101 < l

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-*l* 46.0%

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

      4. associate-*l* 43.6%

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

      5. sub-neg 43.6%

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

      6. +-commutative 43.6%

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

      7. *-commutative 43.6%

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

      8. distribute-rgt-neg-in 43.6%

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

      9. fma-def 43.6%

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

      10. associate-*r/ 50.8%

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

      11. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in l around 0 29.0%

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

      1. pow1/2 30.3%

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

      2. associate-*l* 30.3%

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

   6. Applied egg-rr 30.3%

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

   7. Taylor expanded in n around 0 30.3%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]
   8. Step-by-step derivation

      1. associate-*r* 36.3%

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

   9. Simplified 36.3%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot t\right) \cdot U\right)\right)}}^{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-101}:\\ \;\;\;\;{\left(n \cdot \left(2 \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 17: 35.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. *-commutative 48.7%

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

   2. associate-*l* 49.1%

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

   3. associate-*l* 49.4%

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

   4. associate-*l* 48.2%

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

   5. sub-neg 48.2%

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

   6. +-commutative 48.2%

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

   7. *-commutative 48.2%

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

   8. distribute-rgt-neg-in 48.2%

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

   9. fma-def 48.2%

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

   10. associate-*r/ 54.3%

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

   11. metadata-eval 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in l around 0 39.2%

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

   1. pow1/2 40.4%

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

   2. associate-*l* 40.4%

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

6. Applied egg-rr 40.4%

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

7. Taylor expanded in n around 0 40.4%

   \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]
8. Step-by-step derivation

   1. associate-*r* 39.2%

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

9. Simplified 39.2%

   \[\leadsto {\color{blue}{\left(2 \cdot \left(\left(n \cdot t\right) \cdot U\right)\right)}}^{0.5} \]
10. Step-by-step derivation

    1. *-un-lft-identity 39.2%

       \[\leadsto \color{blue}{1 \cdot {\left(2 \cdot \left(\left(n \cdot t\right) \cdot U\right)\right)}^{0.5}} \]

    2. unpow1/2 36.5%

       \[\leadsto 1 \cdot \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]

    3. associate-*r* 39.2%

       \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(t \cdot U\right)\right)}} \]

11. Applied egg-rr 39.2%

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

    1. *-lft-identity 39.2%

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

    2. associate-*r* 36.5%

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

13. Simplified 36.5%

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

14. Final simplification 36.5%

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

Alternative 18: 35.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

   1. *-commutative 48.7%

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

   2. associate-*l* 49.1%

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

   3. associate-*l* 49.4%

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

   4. associate-*l* 48.2%

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

   5. sub-neg 48.2%

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

   6. +-commutative 48.2%

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

   7. *-commutative 48.2%

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

   8. distribute-rgt-neg-in 48.2%

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

   9. fma-def 48.2%

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

   10. associate-*r/ 54.3%

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

   11. metadata-eval 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in l around 0 39.2%

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

5. Final simplification 39.2%

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

Reproduce

herbie shell --seed 2023175 
(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*))))))