Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 68.4%

Time: 23.9s

Alternatives: 19

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: 49.5% 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: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 10^{-68}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot {\left(\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 1e-68)
     (sqrt
      (*
       2.0
       (* U (* n (+ t (/ l (/ Om (fma -2.0 l (/ n (/ Om (* l U*)))))))))))
     (if (<= t_1 2e+292)
       (sqrt t_1)
       (*
        (* l (sqrt 2.0))
        (pow (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U))))))) 0.5))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 1e-68) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / fma(-2.0, l, (n / (Om / (l * U_42_)))))))))));
	} else if (t_1 <= 2e+292) {
		tmp = sqrt(t_1);
	} else {
		tmp = (l * sqrt(2.0)) * pow((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U))))))), 0.5);
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = 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_42_ - U))))
	tmp = 0.0
	if (t_1 <= 1e-68)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / fma(-2.0, l, Float64(n / Float64(Om / Float64(l * U_42_)))))))))));
	elseif (t_1 <= 2e+292)
		tmp = sqrt(t_1);
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * (Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))))) ^ 0.5));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-68], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(-2.0 * l + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+292], N[Sqrt[t$95$1], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 1.00000000000000007e-68

   1. Initial program 43.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in U around 0 64.5%

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

      1. associate-*r* 68.6%

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

      2. associate-/l* 68.1%

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

      3. fma-def 68.1%

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

      4. associate-/l* 68.1%

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

   6. Simplified 68.1%

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

   if 1.00000000000000007e-68 < (*.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*)))) < 2e292

   1. Initial program 98.9%

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

   if 2e292 < (*.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 22.5%

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

   3. Simplified 36.8%

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

   4. Taylor expanded in t around 0 38.8%

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

   5. Taylor expanded in l around 0 22.9%

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

      1. pow1/2 24.3%

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

      2. associate-/l* 25.6%

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

      3. *-commutative 25.6%

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

      4. sub-neg 25.6%

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

      5. associate-/l* 24.7%

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

      6. metadata-eval 24.7%

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

   7. Applied egg-rr 24.7%

      \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{{\left(\frac{n}{\frac{Om}{U \cdot \left(\frac{n}{\frac{Om}{U* - U}} + -2\right)}}\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-68}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot {\left(\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}\right)}^{0.5}\\ \end{array} \]

Alternative 2: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{n}{\frac{Om}{U* - U}}\\ \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{t_1 - 2}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+240} \lor \neg \left(\ell \leq 5.2 \cdot 10^{+267}\right):\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot {\left(\frac{n}{\frac{Om}{U \cdot \left(-2 + t_1\right)}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ n (/ Om (- U* U)))))
   (if (<= l 4.2e-96)
     (sqrt
      (*
       2.0
       (* U (* n (+ t (/ l (/ Om (fma -2.0 l (/ n (/ Om (* l U*)))))))))))
     (if (<= l 4.8e+113)
       (sqrt (* (* 2.0 n) (+ (* U t) (/ (- t_1 2.0) (/ Om (* U (* l l)))))))
       (if (or (<= l 1.45e+240) (not (<= l 5.2e+267)))
         (* (* l (sqrt 2.0)) (pow (/ n (/ Om (* U (+ -2.0 t_1)))) 0.5))
         (sqrt
          (*
           2.0
           (/ (* (* n l) (* U (fma (* l (/ n Om)) U* (* l -2.0)))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n / (Om / (U_42_ - U));
	double tmp;
	if (l <= 4.2e-96) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / fma(-2.0, l, (n / (Om / (l * U_42_)))))))))));
	} else if (l <= 4.8e+113) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((t_1 - 2.0) / (Om / (U * (l * l)))))));
	} else if ((l <= 1.45e+240) || !(l <= 5.2e+267)) {
		tmp = (l * sqrt(2.0)) * pow((n / (Om / (U * (-2.0 + t_1)))), 0.5);
	} else {
		tmp = sqrt((2.0 * (((n * l) * (U * fma((l * (n / Om)), U_42_, (l * -2.0)))) / Om)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n / Float64(Om / Float64(U_42_ - U)))
	tmp = 0.0
	if (l <= 4.2e-96)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / fma(-2.0, l, Float64(n / Float64(Om / Float64(l * U_42_)))))))))));
	elseif (l <= 4.8e+113)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(t_1 - 2.0) / Float64(Om / Float64(U * Float64(l * l)))))));
	elseif ((l <= 1.45e+240) || !(l <= 5.2e+267))
		tmp = Float64(Float64(l * sqrt(2.0)) * (Float64(n / Float64(Om / Float64(U * Float64(-2.0 + t_1)))) ^ 0.5));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * fma(Float64(l * Float64(n / Om)), U_42_, Float64(l * -2.0)))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.2e-96], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(-2.0 * l + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.8e+113], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(t$95$1 - 2.0), $MachinePrecision] / N[(Om / N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 1.45e+240], N[Not[LessEqual[l, 5.2e+267]], $MachinePrecision]], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n / N[(Om / N[(U * N[(-2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * U$42$ + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 4.20000000000000002e-96

   1. Initial program 51.6%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 59.7%

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

      1. associate-*r* 63.3%

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

      2. associate-/l* 64.2%

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

      3. fma-def 64.2%

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

      4. associate-/l* 64.1%

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

   6. Simplified 64.1%

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

   if 4.20000000000000002e-96 < l < 4.79999999999999966e113

   1. Initial program 53.8%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in l around -inf 61.1%

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

      1. mul-1-neg 61.1%

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

      2. unsub-neg 61.1%

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

      3. *-commutative 61.1%

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

      4. associate-/l* 61.1%

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

      5. mul-1-neg 61.1%

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

      6. unsub-neg 61.1%

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

      7. associate-/l* 63.6%

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

      8. *-commutative 63.6%

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

      9. unpow2 63.6%

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

   6. Simplified 63.6%

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

   if 4.79999999999999966e113 < l < 1.44999999999999999e240 or 5.20000000000000005e267 < l

   1. Initial program 20.3%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in t around 0 38.8%

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

   5. Taylor expanded in l around 0 76.1%

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

      1. pow1/2 76.4%

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

      2. associate-/l* 82.0%

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

      3. *-commutative 82.0%

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

      4. sub-neg 82.0%

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

      5. associate-/l* 78.3%

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

      6. metadata-eval 78.3%

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

   7. Applied egg-rr 78.3%

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

   if 1.44999999999999999e240 < l < 5.20000000000000005e267

   1. Initial program 26.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in U around 0 26.9%

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

      1. associate-*r* 26.9%

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

      2. associate-/l* 26.9%

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

      3. fma-def 26.9%

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

      4. associate-/l* 26.9%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in t around 0 26.9%

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

      1. associate-*r* 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l/ 50.6%

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

      4. +-commutative 50.6%

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

      5. associate-*r* 75.6%

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

      6. fma-def 75.6%

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

      7. *-commutative 75.6%

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

   9. Simplified 75.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{Om}{U* - U}} - 2}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+240} \lor \neg \left(\ell \leq 5.2 \cdot 10^{+267}\right):\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot {\left(\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \]

Alternative 3: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+238} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+268}\right):\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot {\left(\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.02e+152)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (/ n (/ Om (* l U*))) (* l -2.0))) Om)))))
   (if (or (<= l 9.5e+238) (not (<= l 2.1e+268)))
     (*
      (* l (sqrt 2.0))
      (pow (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U))))))) 0.5))
     (sqrt
      (* 2.0 (/ (* (* n l) (* U (fma (* l (/ n Om)) U* (* l -2.0)))) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.02e+152) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	} else if ((l <= 9.5e+238) || !(l <= 2.1e+268)) {
		tmp = (l * sqrt(2.0)) * pow((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U))))))), 0.5);
	} else {
		tmp = sqrt((2.0 * (((n * l) * (U * fma((l * (n / Om)), U_42_, (l * -2.0)))) / Om)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.02e+152)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0))) / Om)))));
	elseif ((l <= 9.5e+238) || !(l <= 2.1e+268))
		tmp = Float64(Float64(l * sqrt(2.0)) * (Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))))) ^ 0.5));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * fma(Float64(l * Float64(n / Om)), U_42_, Float64(l * -2.0)))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.02e+152], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 9.5e+238], N[Not[LessEqual[l, 2.1e+268]], $MachinePrecision]], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * U$42$ + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.01999999999999999e152

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. associate-/l* 59.5%

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

   6. Applied egg-rr 59.5%

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

   if 1.01999999999999999e152 < l < 9.5000000000000003e238 or 2.1000000000000001e268 < l

   1. Initial program 17.7%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in t around 0 41.7%

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

   5. Taylor expanded in l around 0 78.3%

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

      1. pow1/2 78.3%

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

      2. associate-/l* 84.4%

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

      3. *-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. associate-/l* 80.6%

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

      6. metadata-eval 80.6%

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

   7. Applied egg-rr 80.6%

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

   if 9.5000000000000003e238 < l < 2.1000000000000001e268

   1. Initial program 26.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in U around 0 26.9%

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

      1. associate-*r* 26.9%

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

      2. associate-/l* 26.9%

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

      3. fma-def 26.9%

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

      4. associate-/l* 26.9%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in t around 0 26.9%

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

      1. associate-*r* 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l/ 50.6%

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

      4. +-commutative 50.6%

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

      5. associate-*r* 75.6%

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

      6. fma-def 75.6%

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

      7. *-commutative 75.6%

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

   9. Simplified 75.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+238} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+268}\right):\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot {\left(\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \]

Alternative 4: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+238} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+268}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.2e+151)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (/ n (/ Om (* l U*))) (* l -2.0))) Om)))))
   (if (or (<= l 5.1e+238) (not (<= l 3.4e+268)))
     (*
      (sqrt 2.0)
      (* l (sqrt (* n (* U (+ (* (/ n Om) (/ U* Om)) (/ -2.0 Om)))))))
     (sqrt
      (* 2.0 (/ (* (* n l) (* U (fma (* l (/ n Om)) U* (* l -2.0)))) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+151) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	} else if ((l <= 5.1e+238) || !(l <= 3.4e+268)) {
		tmp = sqrt(2.0) * (l * sqrt((n * (U * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om))))));
	} else {
		tmp = sqrt((2.0 * (((n * l) * (U * fma((l * (n / Om)), U_42_, (l * -2.0)))) / Om)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.2e+151)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0))) / Om)))));
	elseif ((l <= 5.1e+238) || !(l <= 3.4e+268))
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) + Float64(-2.0 / Om)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * fma(Float64(l * Float64(n / Om)), U_42_, Float64(l * -2.0)))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.2e+151], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 5.1e+238], N[Not[LessEqual[l, 3.4e+268]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n * N[(U * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * U$42$ + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.2000000000000004e151

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. associate-/l* 59.5%

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

   6. Applied egg-rr 59.5%

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

   if 6.2000000000000004e151 < l < 5.1000000000000002e238 or 3.4000000000000003e268 < l

   1. Initial program 17.7%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in U around 0 31.8%

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

   5. Taylor expanded in l around inf 61.1%

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

      1. associate-*l* 61.0%

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

      2. *-commutative 61.0%

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

      3. sub-neg 61.0%

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

      4. unpow2 61.0%

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

      5. times-frac 76.8%

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

      6. associate-*r/ 76.8%

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

      7. metadata-eval 76.8%

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

      8. distribute-neg-frac 76.8%

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

      9. metadata-eval 76.8%

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

   7. Simplified 76.8%

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

   if 5.1000000000000002e238 < l < 3.4000000000000003e268

   1. Initial program 26.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in U around 0 26.9%

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

      1. associate-*r* 26.9%

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

      2. associate-/l* 26.9%

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

      3. fma-def 26.9%

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

      4. associate-/l* 26.9%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in t around 0 26.9%

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

      1. associate-*r* 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l/ 50.6%

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

      4. +-commutative 50.6%

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

      5. associate-*r* 75.6%

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

      6. fma-def 75.6%

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

      7. *-commutative 75.6%

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

   9. Simplified 75.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+238} \lor \neg \left(\ell \leq 3.4 \cdot 10^{+268}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \]

Alternative 5: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\right)\\ \mathbf{elif}\;\ell \leq 10^{+267}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.75e+152)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (/ n (/ Om (* l U*))) (* l -2.0))) Om)))))
   (if (<= l 1.75e+239)
     (*
      (sqrt 2.0)
      (* l (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U))))))))))
     (if (<= l 1e+267)
       (sqrt
        (* 2.0 (/ (* (* n l) (* U (fma (* l (/ n Om)) U* (* l -2.0)))) Om)))
       (*
        (sqrt 2.0)
        (* l (sqrt (* n (* U (+ (* (/ n Om) (/ U* Om)) (/ -2.0 Om)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.75e+152) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	} else if (l <= 1.75e+239) {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U)))))))));
	} else if (l <= 1e+267) {
		tmp = sqrt((2.0 * (((n * l) * (U * fma((l * (n / Om)), U_42_, (l * -2.0)))) / Om)));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n * (U * (((n / Om) * (U_42_ / Om)) + (-2.0 / Om))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.75e+152)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0))) / Om)))));
	elseif (l <= 1.75e+239)
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))))))));
	elseif (l <= 1e+267)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * fma(Float64(l * Float64(n / Om)), U_42_, Float64(l * -2.0)))) / Om)));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n / Om) * Float64(U_42_ / Om)) + Float64(-2.0 / Om)))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.75e+152], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.75e+239], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * 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]), $MachinePrecision], If[LessEqual[l, 1e+267], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * U$42$ + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n * N[(U * N[(N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.74999999999999991e152

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. associate-/l* 59.5%

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

   6. Applied egg-rr 59.5%

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

   if 1.74999999999999991e152 < l < 1.7500000000000001e239

   1. Initial program 14.7%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in t around 0 33.2%

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

   5. Taylor expanded in l around 0 69.8%

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

      1. associate-*l* 69.7%

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

      2. associate-/l* 75.6%

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

      3. *-commutative 75.6%

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

      4. sub-neg 75.6%

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

      5. associate-/l* 69.8%

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

      6. metadata-eval 69.8%

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

   7. Simplified 69.8%

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

   if 1.7500000000000001e239 < l < 9.9999999999999997e266

   1. Initial program 26.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in U around 0 26.9%

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

      1. associate-*r* 26.9%

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

      2. associate-/l* 26.9%

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

      3. fma-def 26.9%

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

      4. associate-/l* 26.9%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in t around 0 26.9%

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

      1. associate-*r* 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l/ 50.6%

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

      4. +-commutative 50.6%

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

      5. associate-*r* 75.6%

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

      6. fma-def 75.6%

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

      7. *-commutative 75.6%

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

   9. Simplified 75.6%

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

   if 9.9999999999999997e266 < l

   1. Initial program 22.9%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in U around 0 58.7%

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

   5. Taylor expanded in l around inf 88.7%

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

      1. associate-*l* 88.5%

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

      2. *-commutative 88.5%

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

      3. sub-neg 88.5%

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

      4. unpow2 88.5%

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

      5. times-frac 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\right)\\ \mathbf{elif}\;\ell \leq 10^{+267}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 6: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.6e+151)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (/ n (/ Om (* l U*))) (* l -2.0))) Om)))))
   (sqrt (* 2.0 (/ (* (* n l) (* U (fma (* l (/ n Om)) U* (* l -2.0)))) Om)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.6e+151) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	} else {
		tmp = sqrt((2.0 * (((n * l) * (U * fma((l * (n / Om)), U_42_, (l * -2.0)))) / Om)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.6e+151)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0))) / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * fma(Float64(l * Float64(n / Om)), U_42_, Float64(l * -2.0)))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.6e+151], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * U$42$ + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.6000000000000001e151

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. associate-/l* 59.5%

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

   6. Applied egg-rr 59.5%

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

   if 7.6000000000000001e151 < l

   1. Initial program 18.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in U around 0 31.1%

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

      1. associate-*r* 31.1%

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

      2. associate-/l* 37.3%

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

      3. fma-def 37.3%

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

      4. associate-/l* 37.0%

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

   6. Simplified 37.0%

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

   7. Taylor expanded in t around 0 40.6%

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

      1. associate-*r* 43.8%

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

      2. *-commutative 43.8%

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

      3. associate-*l/ 43.4%

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

      4. +-commutative 43.4%

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

      5. associate-*r* 50.1%

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

      6. fma-def 50.1%

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

      7. *-commutative 50.1%

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

   9. Simplified 50.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U*, \ell \cdot -2\right)\right)}{Om}}\\ \end{array} \]

Alternative 7: 57.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e-169)
   (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om))))))))
   (if (<= l 2.8e+152)
     (sqrt
      (* 2.0 (* U (* n (+ t (/ (* l l) (/ Om (+ -2.0 (/ n (/ Om U*))))))))))
     (sqrt
      (*
       2.0
       (*
        (/ 1.0 Om)
        (* (* U l) (* n (+ (* l -2.0) (/ (* n (* l U*)) Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e-169) {
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	} else if (l <= 2.8e+152) {
		tmp = sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))));
	} else {
		tmp = sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * ((l * -2.0) + ((n * (l * U_42_)) / Om)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1d-169) then
        tmp = sqrt((2.0d0 * (u * (n * (t + ((-2.0d0) * (l * (l / om))))))))
    else if (l <= 2.8d+152) then
        tmp = sqrt((2.0d0 * (u * (n * (t + ((l * l) / (om / ((-2.0d0) + (n / (om / u_42))))))))))
    else
        tmp = sqrt((2.0d0 * ((1.0d0 / om) * ((u * l) * (n * ((l * (-2.0d0)) + ((n * (l * u_42)) / om)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e-169) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	} else if (l <= 2.8e+152) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))));
	} else {
		tmp = Math.sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * ((l * -2.0) + ((n * (l * U_42_)) / Om)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1e-169:
		tmp = math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))))
	elif l <= 2.8e+152:
		tmp = math.sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))))
	else:
		tmp = math.sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * ((l * -2.0) + ((n * (l * U_42_)) / Om)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e-169)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	elseif (l <= 2.8e+152)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(l * l) / Float64(Om / Float64(-2.0 + Float64(n / Float64(Om / U_42_))))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(1.0 / Om) * Float64(Float64(U * l) * Float64(n * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1e-169)
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	elseif (l <= 2.8e+152)
		tmp = sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))));
	else
		tmp = sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * ((l * -2.0) + ((n * (l * U_42_)) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e-169], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.8e+152], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(l * l), $MachinePrecision] / N[(Om / N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(1.0 / Om), $MachinePrecision] * N[(N[(U * l), $MachinePrecision] * N[(n * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.00000000000000002e-169

   1. Initial program 51.3%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in U around 0 59.7%

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

      1. associate-*r* 62.5%

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

      2. associate-/l* 63.4%

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

      3. fma-def 63.4%

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

      4. associate-/l* 63.4%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in Om around inf 49.6%

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

      1. unpow2 49.6%

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

      2. associate-*r/ 51.9%

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

   9. Simplified 51.9%

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

   if 1.00000000000000002e-169 < l < 2.8000000000000002e152

   1. Initial program 54.2%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in U around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. associate-/l* 53.8%

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

      3. fma-def 53.8%

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

      4. associate-/l* 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in l around 0 55.3%

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

      1. associate-/l* 53.6%

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

      2. unpow2 53.6%

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

      3. sub-neg 53.6%

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

      4. associate-/l* 53.7%

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

      5. metadata-eval 53.7%

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

   9. Simplified 53.7%

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

   if 2.8000000000000002e152 < l

   1. Initial program 18.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in t around 0 39.7%

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

      1. div-inv 39.7%

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

      2. associate-/l* 39.4%

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

      3. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate-/r/ 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in U* around inf 43.9%

      \[\leadsto \sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 + \color{blue}{\frac{n \cdot \left(\ell \cdot U*\right)}{Om}}\right)\right) \cdot \left(\ell \cdot U\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8: 59.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 - \frac{\left(n \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.2e+151)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
   (sqrt
    (*
     2.0
     (*
      (/ 1.0 Om)
      (* (* n (- (* l -2.0) (/ (* (* n l) (- U U*)) Om))) (* U l)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+151) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.2d+151) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt((2.0d0 * ((1.0d0 / om) * ((n * ((l * (-2.0d0)) - (((n * l) * (u - u_42)) / om))) * (u * l)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+151) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.2e+151:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.2e+151)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(1.0 / Om) * Float64(Float64(n * Float64(Float64(l * -2.0) - Float64(Float64(Float64(n * l) * Float64(U - U_42_)) / Om))) * Float64(U * l)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.2e+151)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.2e+151], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(1.0 / Om), $MachinePrecision] * N[(N[(n * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(N[(n * l), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.2000000000000004e151

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

   if 6.2000000000000004e151 < l

   1. Initial program 18.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in t around 0 39.7%

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

      1. div-inv 39.7%

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

      2. associate-/l* 39.4%

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

      3. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate-/r/ 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in n around 0 43.0%

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

       1. associate-*r* 46.5%

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

   11. Simplified 46.5%

       \[\leadsto \sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 + \color{blue}{\frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}\right)\right) \cdot \left(\ell \cdot U\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 - \frac{\left(n \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 9: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 - \frac{\left(n \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3e+152)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (/ n (/ Om (* l U*))) (* l -2.0))) Om)))))
   (sqrt
    (*
     2.0
     (*
      (/ 1.0 Om)
      (* (* n (- (* l -2.0) (/ (* (* n l) (- U U*)) Om))) (* U l)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e+152) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	} else {
		tmp = sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 <= 3d+152) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((n / (om / (l * u_42))) + (l * (-2.0d0)))) / om)))))
    else
        tmp = sqrt((2.0d0 * ((1.0d0 / om) * ((n * ((l * (-2.0d0)) - (((n * l) * (u - u_42)) / om))) * (u * l)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e+152) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3e+152:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))))
	else:
		tmp = math.sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3e+152)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0))) / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(1.0 / Om) * Float64(Float64(n * Float64(Float64(l * -2.0) - Float64(Float64(Float64(n * l) * Float64(U - U_42_)) / Om))) * Float64(U * l)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3e+152)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((n / (Om / (l * U_42_))) + (l * -2.0))) / Om)))));
	else
		tmp = sqrt((2.0 * ((1.0 / Om) * ((n * ((l * -2.0) - (((n * l) * (U - U_42_)) / Om))) * (U * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3e+152], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(1.0 / Om), $MachinePrecision] * N[(N[(n * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(N[(n * l), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.99999999999999991e152

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. associate-/l* 59.5%

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

   6. Applied egg-rr 59.5%

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

   if 2.99999999999999991e152 < l

   1. Initial program 18.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in t around 0 39.7%

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

      1. div-inv 39.7%

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

      2. associate-/l* 39.4%

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

      3. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate-/r/ 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in n around 0 43.0%

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

       1. associate-*r* 46.5%

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

   11. Simplified 46.5%

       \[\leadsto \sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 + \color{blue}{\frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}}\right)\right) \cdot \left(\ell \cdot U\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 - \frac{\left(n \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot \left(U \cdot \ell\right)\right)\right)}\\ \end{array} \]

Alternative 10: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.8e-164)
   (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om))))))))
   (if (<= l 3.3e+152)
     (sqrt
      (* 2.0 (* U (* n (+ t (/ (* l l) (/ Om (+ -2.0 (/ n (/ Om U*))))))))))
     (sqrt
      (* 2.0 (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l U*)) Om))))) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.8e-164) {
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	} else if (l <= 3.3e+152) {
		tmp = sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))));
	} else {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 <= 6.8d-164) then
        tmp = sqrt((2.0d0 * (u * (n * (t + ((-2.0d0) * (l * (l / om))))))))
    else if (l <= 3.3d+152) then
        tmp = sqrt((2.0d0 * (u * (n * (t + ((l * l) / (om / ((-2.0d0) + (n / (om / u_42))))))))))
    else
        tmp = sqrt((2.0d0 * ((n * (l * (u * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))))) / om)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.8e-164) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	} else if (l <= 3.3e+152) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))));
	} else {
		tmp = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.8e-164:
		tmp = math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))))
	elif l <= 3.3e+152:
		tmp = math.sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))))
	else:
		tmp = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.8e-164)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	elseif (l <= 3.3e+152)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(l * l) / Float64(Om / Float64(-2.0 + Float64(n / Float64(Om / U_42_))))))))));
	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

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.8e-164)
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	elseif (l <= 3.3e+152)
		tmp = sqrt((2.0 * (U * (n * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_))))))))));
	else
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.8e-164], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.3e+152], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(l * l), $MachinePrecision] / N[(Om / N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if l < 6.8e-164

   1. Initial program 51.3%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in U around 0 59.7%

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

      1. associate-*r* 62.5%

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

      2. associate-/l* 63.4%

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

      3. fma-def 63.4%

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

      4. associate-/l* 63.4%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in Om around inf 49.6%

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

      1. unpow2 49.6%

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

      2. associate-*r/ 51.9%

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

   9. Simplified 51.9%

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

   if 6.8e-164 < l < 3.3000000000000001e152

   1. Initial program 54.2%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in U around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. associate-/l* 53.8%

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

      3. fma-def 53.8%

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

      4. associate-/l* 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in l around 0 55.3%

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

      1. associate-/l* 53.6%

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

      2. unpow2 53.6%

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

      3. sub-neg 53.6%

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

      4. associate-/l* 53.7%

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

      5. metadata-eval 53.7%

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

   9. Simplified 53.7%

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

   if 3.3000000000000001e152 < l

   1. Initial program 18.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in U around 0 31.1%

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

      1. associate-*r* 31.1%

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

      2. associate-/l* 37.3%

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

      3. fma-def 37.3%

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

      4. associate-/l* 37.0%

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

   6. Simplified 37.0%

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

   7. Taylor expanded in t around 0 40.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 11: 58.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;\ell \leq 9 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot t_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ (* l -2.0) (/ (* n (* l U*)) Om))))
   (if (<= l 9e+151)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (* l t_1) Om)))))
     (sqrt (* 2.0 (* (/ 1.0 Om) (* (* U l) (* n t_1))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + ((n * (l * U_42_)) / Om);
	double tmp;
	if (l <= 9e+151) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * t_1) / Om)))));
	} else {
		tmp = sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * t_1)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * (-2.0d0)) + ((n * (l * u_42)) / om)
    if (l <= 9d+151) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * t_1) / om)))))
    else
        tmp = sqrt((2.0d0 * ((1.0d0 / om) * ((u * l) * (n * t_1)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + ((n * (l * U_42_)) / Om);
	double tmp;
	if (l <= 9e+151) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * t_1) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * t_1)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = (l * -2.0) + ((n * (l * U_42_)) / Om)
	tmp = 0
	if l <= 9e+151:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * t_1) / Om)))))
	else:
		tmp = math.sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * t_1)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))
	tmp = 0.0
	if (l <= 9e+151)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * t_1) / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(1.0 / Om) * Float64(Float64(U * l) * Float64(n * t_1)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * -2.0) + ((n * (l * U_42_)) / Om);
	tmp = 0.0;
	if (l <= 9e+151)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * t_1) / Om)))));
	else
		tmp = sqrt((2.0 * ((1.0 / Om) * ((U * l) * (n * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 9e+151], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(1.0 / Om), $MachinePrecision] * N[(N[(U * l), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 8.9999999999999997e151

   1. Initial program 52.0%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in U around 0 58.2%

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

   if 8.9999999999999997e151 < l

   1. Initial program 18.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in t around 0 39.7%

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

      1. div-inv 39.7%

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

      2. associate-/l* 39.4%

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

      3. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. *-commutative 39.4%

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

      2. associate-*r* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate-/r/ 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in U* around inf 43.9%

      \[\leadsto \sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \left(\ell \cdot -2 + \color{blue}{\frac{n \cdot \left(\ell \cdot U*\right)}{Om}}\right)\right) \cdot \left(\ell \cdot U\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{1}{Om} \cdot \left(\left(U \cdot \ell\right) \cdot \left(n \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12: 53.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-104} \lor \neg \left(n \leq 1.6 \cdot 10^{-69}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -3.8e-104) (not (<= n 1.6e-69)))
   (sqrt (* (* 2.0 n) (* U (+ t (* (/ n Om) (/ (* (* l l) U*) Om))))))
   (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -3.8e-104) || !(n <= 1.6e-69)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((n / Om) * (((l * l) * U_42_) / Om))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-3.8d-104)) .or. (.not. (n <= 1.6d-69))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((n / om) * (((l * l) * u_42) / om))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + ((-2.0d0) * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -3.8e-104) || !(n <= 1.6e-69)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((n / Om) * (((l * l) * U_42_) / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -3.8e-104) or not (n <= 1.6e-69):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((n / Om) * (((l * l) * U_42_) / Om))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -3.8e-104) || !(n <= 1.6e-69))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(n / Om) * Float64(Float64(Float64(l * l) * U_42_) / Om))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -3.8e-104) || ~((n <= 1.6e-69)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((n / Om) * (((l * l) * U_42_) / Om))))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -3.8e-104], N[Not[LessEqual[n, 1.6e-69]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.8000000000000001e-104 or 1.59999999999999999e-69 < n

   1. Initial program 56.1%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in U around 0 60.5%

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

   5. Taylor expanded in n around inf 51.1%

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

      1. unpow2 51.1%

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

      2. times-frac 55.1%

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

      3. *-commutative 55.1%

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

      4. unpow2 55.1%

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

   7. Simplified 55.1%

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

   if -3.8000000000000001e-104 < n < 1.59999999999999999e-69

   1. Initial program 36.6%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in U around 0 47.2%

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

      1. associate-*r* 54.7%

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

      2. associate-/l* 59.2%

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

      3. fma-def 59.2%

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

      4. associate-/l* 59.1%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in Om around inf 50.0%

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

      1. unpow2 50.0%

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

      2. associate-*r/ 56.4%

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

   9. Simplified 56.4%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-104} \lor \neg \left(n \leq 1.6 \cdot 10^{-69}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13: 54.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-103} \lor \neg \left(n \leq 5.1 \cdot 10^{-76}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{U*}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.4e-103) (not (<= n 5.1e-76)))
   (sqrt (* (* (* 2.0 n) U) (+ t (* (/ n Om) (/ U* (/ Om (* l l)))))))
   (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.4e-103) || !(n <= 5.1e-76)) {
		tmp = sqrt((((2.0 * n) * U) * (t + ((n / Om) * (U_42_ / (Om / (l * l)))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-1.4d-103)) .or. (.not. (n <= 5.1d-76))) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((n / om) * (u_42 / (om / (l * l)))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + ((-2.0d0) * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.4e-103) || !(n <= 5.1e-76)) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + ((n / Om) * (U_42_ / (Om / (l * l)))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.4e-103) or not (n <= 5.1e-76):
		tmp = math.sqrt((((2.0 * n) * U) * (t + ((n / Om) * (U_42_ / (Om / (l * l)))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.4e-103) || !(n <= 5.1e-76))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(n / Om) * Float64(U_42_ / Float64(Om / Float64(l * l)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.4e-103) || ~((n <= 5.1e-76)))
		tmp = sqrt((((2.0 * n) * U) * (t + ((n / Om) * (U_42_ / (Om / (l * l)))))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.4e-103], N[Not[LessEqual[n, 5.1e-76]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.40000000000000011e-103 or 5.09999999999999986e-76 < n

   1. Initial program 55.8%

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

   3. Simplified 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 U around 0 60.1%

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

   5. Taylor expanded in n around inf 50.8%

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

      1. unpow2 50.8%

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

      2. times-frac 54.8%

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

      3. *-commutative 54.8%

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

      4. unpow2 54.8%

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

   7. Simplified 54.8%

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

      1. *-un-lft-identity 54.8%

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

      2. *-commutative 54.8%

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

   9. Applied egg-rr 54.8%

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

       1. *-lft-identity 54.8%

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

       2. associate-*r* 55.7%

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

       3. unpow2 55.7%

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

       4. associate-/l* 57.0%

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

       5. unpow2 57.0%

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

   11. Simplified 57.0%

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

   if -1.40000000000000011e-103 < n < 5.09999999999999986e-76

   1. Initial program 36.6%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in U around 0 47.5%

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

      1. associate-*r* 55.2%

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

      2. associate-/l* 59.9%

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

      3. fma-def 59.9%

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

      4. associate-/l* 59.7%

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

   6. Simplified 59.7%

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

   7. Taylor expanded in Om around inf 50.4%

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

      1. unpow2 50.4%

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

      2. associate-*r/ 57.0%

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

   9. Simplified 57.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-103} \lor \neg \left(n \leq 5.1 \cdot 10^{-76}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{n}{Om} \cdot \frac{U*}{\frac{Om}{\ell \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 14: 48.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq -2.3 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -2.3e+166)
   (sqrt (* 2.0 (/ (* n (/ (* (* n (* l l)) (* U U*)) Om)) Om)))
   (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.3e+166) {
		tmp = sqrt((2.0 * ((n * (((n * (l * l)) * (U * U_42_)) / Om)) / Om)));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= (-2.3d+166)) then
        tmp = sqrt((2.0d0 * ((n * (((n * (l * l)) * (u * u_42)) / om)) / om)))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + ((-2.0d0) * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.3e+166) {
		tmp = Math.sqrt((2.0 * ((n * (((n * (l * l)) * (U * U_42_)) / Om)) / Om)));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -2.3e+166:
		tmp = math.sqrt((2.0 * ((n * (((n * (l * l)) * (U * U_42_)) / Om)) / Om)))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -2.3e+166)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(Float64(Float64(n * Float64(l * l)) * Float64(U * U_42_)) / Om)) / Om)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -2.3e+166)
		tmp = sqrt((2.0 * ((n * (((n * (l * l)) * (U * U_42_)) / Om)) / Om)));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -2.3e+166], N[Sqrt[N[(2.0 * N[(N[(n * N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.30000000000000008e166

   1. Initial program 49.0%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in t around 0 41.7%

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

   5. Taylor expanded in U* around inf 41.4%

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

      1. associate-*r* 41.3%

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

      2. unpow2 41.3%

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

   7. Simplified 41.3%

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

   if -2.30000000000000008e166 < U*

   1. Initial program 48.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in U around 0 54.3%

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

      1. associate-*r* 55.6%

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

      2. associate-/l* 57.1%

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

      3. fma-def 57.1%

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

      4. associate-/l* 57.1%

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

   6. Simplified 57.1%

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

   7. Taylor expanded in Om around inf 47.2%

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

      1. unpow2 47.2%

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

      2. associate-*r/ 50.6%

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

   9. Simplified 50.6%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.3 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om}}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 15: 48.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq -2.3 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -2.3e+166)
   (sqrt (* 2.0 (/ (* (* (* n l) (* n l)) (* U U*)) (* Om Om))))
   (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.3e+166) {
		tmp = sqrt((2.0 * ((((n * l) * (n * l)) * (U * U_42_)) / (Om * Om))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= (-2.3d+166)) then
        tmp = sqrt((2.0d0 * ((((n * l) * (n * l)) * (u * u_42)) / (om * om))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + ((-2.0d0) * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.3e+166) {
		tmp = Math.sqrt((2.0 * ((((n * l) * (n * l)) * (U * U_42_)) / (Om * Om))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -2.3e+166:
		tmp = math.sqrt((2.0 * ((((n * l) * (n * l)) * (U * U_42_)) / (Om * Om))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -2.3e+166)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(n * l) * Float64(n * l)) * Float64(U * U_42_)) / Float64(Om * Om))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -2.3e+166)
		tmp = sqrt((2.0 * ((((n * l) * (n * l)) * (U * U_42_)) / (Om * Om))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (-2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -2.3e+166], N[Sqrt[N[(2.0 * N[(N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.30000000000000008e166

   1. Initial program 49.0%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in U around 0 60.6%

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

      1. *-un-lft-identity 60.6%

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

      2. associate-/l* 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in n around inf 34.6%

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

      1. *-commutative 34.6%

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

      2. associate-*r* 34.6%

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

      3. unpow2 34.6%

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

      4. unpow2 34.6%

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

      5. swap-sqr 46.9%

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

      6. *-commutative 46.9%

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

      7. unpow2 46.9%

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

   9. Simplified 46.9%

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

   if -2.30000000000000008e166 < U*

   1. Initial program 48.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in U around 0 54.3%

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

      1. associate-*r* 55.6%

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

      2. associate-/l* 57.1%

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

      3. fma-def 57.1%

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

      4. associate-/l* 57.1%

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

   6. Simplified 57.1%

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

   7. Taylor expanded in Om around inf 47.2%

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

      1. unpow2 47.2%

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

      2. associate-*r/ 50.6%

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

   9. Simplified 50.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.3 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot U*\right)}{Om \cdot Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 16: 48.8% accurate, 2.0× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* 2.0 (* U (* n (+ t (* -2.0 (* l (/ l Om)))))))))

C

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

Fortran

NOTE: l should be positive before calling this function
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 + ((-2.0d0) * (l * (l / om))))))))
end function

Java

l = Math.abs(l);
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 + (-2.0 * (l * (l / Om))))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Simplified 54.6%

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

4. Taylor expanded in U around 0 55.2%

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

   1. associate-*r* 57.1%

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

   2. associate-/l* 58.4%

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

   3. fma-def 58.4%

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

   4. associate-/l* 58.3%

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

6. Simplified 58.3%

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

7. Taylor expanded in Om around inf 44.8%

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

   1. unpow2 44.8%

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

   2. associate-*r/ 47.8%

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

9. Simplified 47.8%

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

10. Final simplification 47.8%

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

Alternative 17: 37.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 2.7 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 2.7e-149)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 2.7e-149) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 2.7d-149) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 2.7e-149) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 2.7e-149:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 2.7e-149)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 2.7e-149)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 2.7e-149], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.70000000000000014e-149

   1. Initial program 48.3%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in U around 0 53.4%

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

      1. associate-*r* 58.4%

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

      2. associate-/l* 61.2%

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

      3. fma-def 61.2%

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

      4. associate-/l* 61.2%

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

   6. Simplified 61.2%

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

   7. Taylor expanded in t around inf 35.4%

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

      1. associate-*r* 41.1%

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

   9. Simplified 41.1%

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

   if 2.70000000000000014e-149 < n

   1. Initial program 48.2%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in t around inf 37.8%

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

      1. pow1/2 40.8%

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

      2. associate-*l* 40.8%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.7 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 18: 38.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* U (* n t))) 0.5))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (U * (n * t))), 0.5);
}

Fortran

NOTE: l should be positive before calling this function
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 * (u * (n * t))) ** 0.5d0
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((2.0 * (U * (n * t))), 0.5);
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (U * (n * t))), 0.5)

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5
end

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 48.3%

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

3. Simplified 54.6%

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

4. Taylor expanded in U around 0 55.2%

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

   1. associate-*r* 57.1%

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

   2. associate-/l* 58.4%

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

   3. fma-def 58.4%

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

   4. associate-/l* 58.3%

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

6. Simplified 58.3%

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

7. Taylor expanded in t around inf 36.4%

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

   1. associate-*r* 38.7%

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

9. Simplified 38.7%

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

    1. pow1/2 40.3%

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

    2. associate-*r* 38.0%

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

    3. associate-*r* 38.0%

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

11. Applied egg-rr 38.0%

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

12. Taylor expanded in n around 0 38.0%

    \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}}^{0.5} \]
13. Step-by-step derivation

    1. associate-*r* 40.3%

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

14. Simplified 40.3%

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

15. Final simplification 40.3%

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

Alternative 19: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

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

Fortran

NOTE: l should be positive before calling this function
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

l = Math.abs(l);
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

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
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 48.3%

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

3. Simplified 54.6%

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

4. Taylor expanded in U around 0 55.2%

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

   1. associate-*r* 57.1%

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

   2. associate-/l* 58.4%

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

   3. fma-def 58.4%

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

   4. associate-/l* 58.3%

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

6. Simplified 58.3%

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

7. Taylor expanded in t around inf 36.4%

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

   1. associate-*r* 38.7%

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

9. Simplified 38.7%

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

10. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023217 
(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*))))))