Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 64.7%

Time: 34.6s

Alternatives: 18

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 5e-27)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
     (if (<= t_1 2e+305)
       (sqrt t_1)
       (*
        (* l (sqrt 2.0))
        (sqrt (* n (* U (- (/ n (/ (* Om Om) (- U* U))) (/ 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 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 5e-27) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U)))))))));
	} else if (t_1 <= 2e+305) {
		tmp = sqrt(t_1);
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * ((n / ((Om * Om) / (U_42_ - U))) - (2.0 / Om)))));
	}
	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 <= 5e-27)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	elseif (t_1 <= 2e+305)
		tmp = sqrt(t_1);
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n * Float64(U * Float64(Float64(n / Float64(Float64(Om * Om) / Float64(U_42_ - U))) - 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$42$_] := 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, 5e-27], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[Sqrt[t$95$1], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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*)))) < 5.0000000000000002e-27

   1. Initial program 45.7%

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

   3. Simplified 69.0%

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

   if 5.0000000000000002e-27 < (*.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*)))) < 1.9999999999999999e305

   1. Initial program 99.6%

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

   if 1.9999999999999999e305 < (*.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 26.2%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in l around inf 23.5%

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

      1. associate-/l* 23.8%

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

      2. unpow2 23.8%

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

      3. associate-*r/ 23.8%

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

      4. metadata-eval 23.8%

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

   6. Simplified 23.8%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 2: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (/ (* n (* l U*)) Om) (* l -2.0))) Om)))))))
   (if (<= l 6.5e+77)
     t_1
     (if (<= l 2.8e+130)
       (sqrt (* (* -2.0 (* (* l l) (/ n Om))) (* U (- 2.0 (/ (* n U*) Om)))))
       (if (<= l 2.3e+145)
         t_1
         (*
          (* l (sqrt 2.0))
          (sqrt (* n (* U (+ (/ (* n (- U* U)) (* Om Om)) (/ -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 = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	double tmp;
	if (l <= 6.5e+77) {
		tmp = t_1;
	} else if (l <= 2.8e+130) {
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else if (l <= 2.3e+145) {
		tmp = t_1;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * (((n * (U_42_ - U)) / (Om * Om)) + (-2.0 / 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * (u * (t + ((l * (((n * (l * u_42)) / om) + (l * (-2.0d0)))) / om)))))
    if (l <= 6.5d+77) then
        tmp = t_1
    else if (l <= 2.8d+130) then
        tmp = sqrt((((-2.0d0) * ((l * l) * (n / om))) * (u * (2.0d0 - ((n * u_42) / om)))))
    else if (l <= 2.3d+145) then
        tmp = t_1
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n * (u * (((n * (u_42 - u)) / (om * om)) + ((-2.0d0) / 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 t_1 = Math.sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	double tmp;
	if (l <= 6.5e+77) {
		tmp = t_1;
	} else if (l <= 2.8e+130) {
		tmp = Math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else if (l <= 2.3e+145) {
		tmp = t_1;
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n * (U * (((n * (U_42_ - U)) / (Om * Om)) + (-2.0 / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))))
	tmp = 0
	if l <= 6.5e+77:
		tmp = t_1
	elif l <= 2.8e+130:
		tmp = math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))))
	elif l <= 2.3e+145:
		tmp = t_1
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n * (U * (((n * (U_42_ - U)) / (Om * Om)) + (-2.0 / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))) / Om)))))
	tmp = 0.0
	if (l <= 6.5e+77)
		tmp = t_1;
	elseif (l <= 2.8e+130)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l * l) * Float64(n / Om))) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))));
	elseif (l <= 2.3e+145)
		tmp = t_1;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) + Float64(-2.0 / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	tmp = 0.0;
	if (l <= 6.5e+77)
		tmp = t_1;
	elseif (l <= 2.8e+130)
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	elseif (l <= 2.3e+145)
		tmp = t_1;
	else
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * (((n * (U_42_ - U)) / (Om * Om)) + (-2.0 / Om)))));
	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[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 6.5e+77], t$95$1, If[LessEqual[l, 2.8e+130], N[Sqrt[N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.3e+145], t$95$1, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.5e77 or 2.7999999999999999e130 < l < 2.3e145

   1. Initial program 54.5%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in U around 0 60.1%

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

   if 6.5e77 < l < 2.7999999999999999e130

   1. Initial program 41.0%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in l around -inf 68.2%

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

      1. associate-/l* 36.6%

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

      2. associate-/r* 37.0%

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

      3. unpow2 37.0%

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

      4. *-commutative 37.0%

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

      5. mul-1-neg 37.0%

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

      6. unsub-neg 37.0%

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

      7. associate-/l* 37.0%

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

   6. Simplified 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. associate-/r/ 37.0%

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

      3. associate-/r/ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. *-lft-identity 37.0%

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

      2. associate-*r* 37.7%

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

      3. unpow2 37.7%

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

      4. associate-/r/ 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*l/ 62.8%

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

      7. associate-*l/ 62.8%

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

      8. *-commutative 62.8%

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

   10. Simplified 62.8%

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

   11. Taylor expanded in U* around inf 62.8%

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

   if 2.3e145 < l

   1. Initial program 9.3%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in l around inf 69.6%

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

      1. sub-neg 69.6%

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

      2. unpow2 69.6%

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

      3. associate-*r/ 69.6%

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

      4. metadata-eval 69.6%

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

      5. distribute-neg-frac 69.6%

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

      6. metadata-eval 69.6%

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

   6. Simplified 69.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} + \frac{-2}{Om}\right)\right)}\\ \end{array} \]

Alternative 3: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (/ (* n (* l U*)) Om) (* l -2.0))) Om)))))))
   (if (<= l 4.5e+78)
     t_1
     (if (<= l 3.6e+130)
       (sqrt (* (* -2.0 (* (* l l) (/ n Om))) (* U (- 2.0 (/ (* n U*) Om)))))
       (if (<= l 2.4e+145)
         t_1
         (*
          (* l (sqrt 2.0))
          (sqrt (* n (* U (- (/ n (/ (* Om Om) (- U* U))) (/ 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 = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	double tmp;
	if (l <= 4.5e+78) {
		tmp = t_1;
	} else if (l <= 3.6e+130) {
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else if (l <= 2.4e+145) {
		tmp = t_1;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * ((n / ((Om * Om) / (U_42_ - U))) - (2.0 / 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * (u * (t + ((l * (((n * (l * u_42)) / om) + (l * (-2.0d0)))) / om)))))
    if (l <= 4.5d+78) then
        tmp = t_1
    else if (l <= 3.6d+130) then
        tmp = sqrt((((-2.0d0) * ((l * l) * (n / om))) * (u * (2.0d0 - ((n * u_42) / om)))))
    else if (l <= 2.4d+145) then
        tmp = t_1
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((n * (u * ((n / ((om * om) / (u_42 - u))) - (2.0d0 / 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 t_1 = Math.sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	double tmp;
	if (l <= 4.5e+78) {
		tmp = t_1;
	} else if (l <= 3.6e+130) {
		tmp = Math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else if (l <= 2.4e+145) {
		tmp = t_1;
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((n * (U * ((n / ((Om * Om) / (U_42_ - U))) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))))
	tmp = 0
	if l <= 4.5e+78:
		tmp = t_1
	elif l <= 3.6e+130:
		tmp = math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))))
	elif l <= 2.4e+145:
		tmp = t_1
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((n * (U * ((n / ((Om * Om) / (U_42_ - U))) - (2.0 / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))) / Om)))))
	tmp = 0.0
	if (l <= 4.5e+78)
		tmp = t_1;
	elseif (l <= 3.6e+130)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l * l) * Float64(n / Om))) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))));
	elseif (l <= 2.4e+145)
		tmp = t_1;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n * Float64(U * Float64(Float64(n / Float64(Float64(Om * Om) / Float64(U_42_ - U))) - Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	tmp = 0.0;
	if (l <= 4.5e+78)
		tmp = t_1;
	elseif (l <= 3.6e+130)
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	elseif (l <= 2.4e+145)
		tmp = t_1;
	else
		tmp = (l * sqrt(2.0)) * sqrt((n * (U * ((n / ((Om * Om) / (U_42_ - U))) - (2.0 / Om)))));
	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[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 4.5e+78], t$95$1, If[LessEqual[l, 3.6e+130], N[Sqrt[N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.4e+145], t$95$1, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(U * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.4999999999999999e78 or 3.6000000000000001e130 < l < 2.39999999999999992e145

   1. Initial program 54.5%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in U around 0 60.1%

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

   if 4.4999999999999999e78 < l < 3.6000000000000001e130

   1. Initial program 41.0%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in l around -inf 68.2%

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

      1. associate-/l* 36.6%

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

      2. associate-/r* 37.0%

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

      3. unpow2 37.0%

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

      4. *-commutative 37.0%

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

      5. mul-1-neg 37.0%

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

      6. unsub-neg 37.0%

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

      7. associate-/l* 37.0%

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

   6. Simplified 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. associate-/r/ 37.0%

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

      3. associate-/r/ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. *-lft-identity 37.0%

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

      2. associate-*r* 37.7%

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

      3. unpow2 37.7%

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

      4. associate-/r/ 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*l/ 62.8%

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

      7. associate-*l/ 62.8%

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

      8. *-commutative 62.8%

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

   10. Simplified 62.8%

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

   11. Taylor expanded in U* around inf 62.8%

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

   if 2.39999999999999992e145 < l

   1. Initial program 9.3%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in l around inf 69.6%

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

      1. associate-/l* 70.4%

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

      2. unpow2 70.4%

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

      3. associate-*r/ 70.4%

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

      4. metadata-eval 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 4: 56.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 0.005:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 0.005)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
   (sqrt (* (* (* 2.0 n) U) (+ 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 <= 0.005) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U)))))))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (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 <= 0.005)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / 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[U, 0.005], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 0.0050000000000000001

   1. Initial program 47.0%

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

   3. Simplified 62.5%

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

   if 0.0050000000000000001 < U

   1. Initial program 59.7%

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

   2. Taylor expanded in Om around inf 54.4%

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

      1. unpow2 54.4%

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

      2. associate-*r/ 59.9%

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

   4. Simplified 59.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 0.005:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 5: 51.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq 1.12 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+233}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))))
   (if (<= l 1.12e-157)
     (sqrt (* (* 2.0 n) (* U t)))
     (if (<= l 1.25e+32)
       (sqrt (* (* 2.0 n) (* U (+ t (* (/ (* n (* l l)) Om) (/ U* Om))))))
       (if (<= l 6.2e+65)
         t_1
         (if (<= l 9.2e+170)
           (sqrt
            (* (* -2.0 (* (* l l) (/ n Om))) (* U (- 2.0 (/ (* n U*) Om)))))
           (if (<= l 3.8e+233)
             t_1
             (sqrt
              (*
               -2.0
               (/
                n
                (/
                 (/ (/ Om l) l)
                 (* U (- 2.0 (/ n (/ Om (- U* U))))))))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	double tmp;
	if (l <= 1.12e-157) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 1.25e+32) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else if (l <= 6.2e+65) {
		tmp = t_1;
	} else if (l <= 9.2e+170) {
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else if (l <= 3.8e+233) {
		tmp = t_1;
	} else {
		tmp = sqrt((-2.0 * (n / (((Om / l) / l) / (U * (2.0 - (n / (Om / (U_42_ - U)))))))));
	}
	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 = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    if (l <= 1.12d-157) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else if (l <= 1.25d+32) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((n * (l * l)) / om) * (u_42 / om))))))
    else if (l <= 6.2d+65) then
        tmp = t_1
    else if (l <= 9.2d+170) then
        tmp = sqrt((((-2.0d0) * ((l * l) * (n / om))) * (u * (2.0d0 - ((n * u_42) / om)))))
    else if (l <= 3.8d+233) then
        tmp = t_1
    else
        tmp = sqrt(((-2.0d0) * (n / (((om / l) / l) / (u * (2.0d0 - (n / (om / (u_42 - u)))))))))
    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 = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	double tmp;
	if (l <= 1.12e-157) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 1.25e+32) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else if (l <= 6.2e+65) {
		tmp = t_1;
	} else if (l <= 9.2e+170) {
		tmp = Math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else if (l <= 3.8e+233) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((-2.0 * (n / (((Om / l) / l) / (U * (2.0 - (n / (Om / (U_42_ - U)))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	tmp = 0
	if l <= 1.12e-157:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	elif l <= 1.25e+32:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))))
	elif l <= 6.2e+65:
		tmp = t_1
	elif l <= 9.2e+170:
		tmp = math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))))
	elif l <= 3.8e+233:
		tmp = t_1
	else:
		tmp = math.sqrt((-2.0 * (n / (((Om / l) / l) / (U * (2.0 - (n / (Om / (U_42_ - U)))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))
	tmp = 0.0
	if (l <= 1.12e-157)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (l <= 1.25e+32)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(n * Float64(l * l)) / Om) * Float64(U_42_ / Om))))));
	elseif (l <= 6.2e+65)
		tmp = t_1;
	elseif (l <= 9.2e+170)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l * l) * Float64(n / Om))) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))));
	elseif (l <= 3.8e+233)
		tmp = t_1;
	else
		tmp = sqrt(Float64(-2.0 * Float64(n / Float64(Float64(Float64(Om / l) / l) / Float64(U * Float64(2.0 - Float64(n / Float64(Om / Float64(U_42_ - U)))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	tmp = 0.0;
	if (l <= 1.12e-157)
		tmp = sqrt(((2.0 * n) * (U * t)));
	elseif (l <= 1.25e+32)
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	elseif (l <= 6.2e+65)
		tmp = t_1;
	elseif (l <= 9.2e+170)
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	elseif (l <= 3.8e+233)
		tmp = t_1;
	else
		tmp = sqrt((-2.0 * (n / (((Om / l) / l) / (U * (2.0 - (n / (Om / (U_42_ - U)))))))));
	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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1.12e-157], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.25e+32], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.2e+65], t$95$1, If[LessEqual[l, 9.2e+170], N[Sqrt[N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e+233], t$95$1, N[Sqrt[N[(-2.0 * N[(n / N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] / N[(U * N[(2.0 - N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 1.12000000000000001e-157

   1. Initial program 55.1%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in t around inf 40.0%

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

   if 1.12000000000000001e-157 < l < 1.2499999999999999e32

   1. Initial program 46.9%

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

      1. associate-*l* 53.5%

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

      2. sub-neg 53.5%

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

      3. associate-+l- 53.5%

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

      4. sub-neg 53.5%

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

      5. associate-/l* 53.5%

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

      6. remove-double-neg 53.5%

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

      7. associate-*l* 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in U* around inf 48.2%

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

      1. mul-1-neg 48.2%

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

      2. associate-/l* 45.5%

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

      3. distribute-neg-frac 45.5%

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

      4. *-commutative 45.5%

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

      5. associate-/r* 45.6%

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

      6. unpow2 45.6%

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

      7. unpow2 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in U around 0 48.2%

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

      1. associate-*r* 48.2%

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

      2. *-commutative 48.2%

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

      3. sub-neg 48.2%

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

      4. mul-1-neg 48.2%

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

      5. remove-double-neg 48.2%

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

      6. associate-*r* 48.2%

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

      7. unpow2 48.2%

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

      8. times-frac 50.9%

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

      9. unpow2 50.9%

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

   9. Simplified 50.9%

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

   if 1.2499999999999999e32 < l < 6.19999999999999981e65 or 9.2000000000000003e170 < l < 3.7999999999999999e233

   1. Initial program 29.8%

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

   2. Taylor expanded in Om around inf 30.4%

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

      1. unpow2 30.4%

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

      2. associate-*r/ 57.1%

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

   4. Simplified 57.1%

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

   if 6.19999999999999981e65 < l < 9.2000000000000003e170

   1. Initial program 37.5%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in l around -inf 62.1%

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

      1. associate-/l* 43.1%

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

      2. associate-/r* 43.2%

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

      3. unpow2 43.2%

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

      4. *-commutative 43.2%

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

      5. mul-1-neg 43.2%

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

      6. unsub-neg 43.2%

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

      7. associate-/l* 43.2%

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

   6. Simplified 43.2%

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

      1. *-un-lft-identity 43.2%

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

      2. associate-/r/ 49.6%

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

      3. associate-/r/ 49.6%

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

   8. Applied egg-rr 49.6%

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

      1. *-lft-identity 49.6%

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

      2. associate-*r* 49.9%

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

      3. unpow2 49.9%

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

      4. associate-/r/ 59.9%

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

      5. unpow2 59.9%

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

      6. associate-*l/ 59.9%

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

      7. associate-*l/ 59.9%

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

      8. *-commutative 59.9%

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

   10. Simplified 59.9%

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

   11. Taylor expanded in U* around inf 59.9%

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

   if 3.7999999999999999e233 < l

   1. Initial program 14.2%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in l around -inf 53.8%

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

      1. associate-/l* 53.8%

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

      2. associate-/r* 53.8%

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

      3. unpow2 53.8%

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

      4. *-commutative 53.8%

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

      5. mul-1-neg 53.8%

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

      6. unsub-neg 53.8%

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

      7. associate-/l* 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in Om around 0 53.8%

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

      1. unpow2 53.8%

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

      2. associate-/r* 53.8%

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

   9. Simplified 53.8%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U \cdot \left(2 - \frac{n}{\frac{Om}{U* - U}}\right)}}}\\ \end{array} \]

Alternative 6: 57.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.34 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.34e+125)
   (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (* l (/ l Om))))))
   (if (<= Om 5.6e+197)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (/ (* n (* l U*)) Om) (* l -2.0))) Om)))))
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ l (/ Om (- (* l -2.0) (/ n (/ 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 (Om <= -1.34e+125) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else if (Om <= 5.6e+197) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (l / (Om / ((l * -2.0) - (n / (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 (om <= (-1.34d+125)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l * (l / om))))))
    else if (om <= 5.6d+197) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * (((n * (l * u_42)) / om) + (l * (-2.0d0)))) / om)))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + (l / (om / ((l * (-2.0d0)) - (n / (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 (Om <= -1.34e+125) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	} else if (Om <= 5.6e+197) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l))))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1.34e+125:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))))
	elif Om <= 5.6e+197:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l))))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.34e+125)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))));
	elseif (Om <= 5.6e+197)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(n / Float64(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 (Om <= -1.34e+125)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l * (l / Om))))));
	elseif (Om <= 5.6e+197)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (l / (Om / ((l * -2.0) - (n / (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[Om, -1.34e+125], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 5.6e+197], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(n / N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.3399999999999999e125

   1. Initial program 42.5%

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

   2. Taylor expanded in Om around inf 42.5%

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

      1. unpow2 42.5%

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

      2. associate-*r/ 63.4%

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

   4. Simplified 63.4%

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

   if -1.3399999999999999e125 < Om < 5.5999999999999997e197

   1. Initial program 52.2%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in U around 0 61.0%

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

   if 5.5999999999999997e197 < Om

   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 49.5%

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

   4. Taylor expanded in U* around 0 56.2%

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

      1. *-commutative 56.2%

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

      2. +-commutative 56.2%

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

      3. associate-/l* 62.8%

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

      4. +-commutative 62.8%

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

      5. mul-1-neg 62.8%

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

      6. unsub-neg 62.8%

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

      7. *-commutative 62.8%

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

      8. associate-/l* 66.2%

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

   6. Simplified 66.2%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.34 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)}\\ \end{array} \]

Alternative 7: 51.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2
         (sqrt (* (* 2.0 n) (* U (+ t (* (/ (* n (* l l)) Om) (/ U* Om))))))))
   (if (<= n -1.12e-57)
     t_2
     (if (<= n -2.35e-283)
       (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1)))))
       (if (<= n 7e-59) (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 t_1)))) t_2)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	double tmp;
	if (n <= -1.12e-57) {
		tmp = t_2;
	} else if (n <= -2.35e-283) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else if (n <= 7e-59) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = l * (l / om)
    t_2 = sqrt(((2.0d0 * n) * (u * (t + (((n * (l * l)) / om) * (u_42 / om))))))
    if (n <= (-1.12d-57)) then
        tmp = t_2
    else if (n <= (-2.35d-283)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else if (n <= 7d-59) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * t_1))))
    else
        tmp = t_2
    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 * (l / Om);
	double t_2 = Math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	double tmp;
	if (n <= -1.12e-57) {
		tmp = t_2;
	} else if (n <= -2.35e-283) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else if (n <= 7e-59) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))))
	tmp = 0
	if n <= -1.12e-57:
		tmp = t_2
	elif n <= -2.35e-283:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	elif n <= 7e-59:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))))
	else:
		tmp = t_2
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(n * Float64(l * l)) / Om) * Float64(U_42_ / Om))))))
	tmp = 0.0
	if (n <= -1.12e-57)
		tmp = t_2;
	elseif (n <= -2.35e-283)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	elseif (n <= 7e-59)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	tmp = 0.0;
	if (n <= -1.12e-57)
		tmp = t_2;
	elseif (n <= -2.35e-283)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	elseif (n <= 7e-59)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	else
		tmp = t_2;
	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[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.12e-57], t$95$2, If[LessEqual[n, -2.35e-283], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 7e-59], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.12e-57 or 7.0000000000000002e-59 < n

   1. Initial program 55.0%

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

      1. associate-*l* 57.4%

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

      2. sub-neg 57.4%

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

      3. associate-+l- 57.4%

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

      4. sub-neg 57.4%

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

      5. associate-/l* 60.9%

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

      6. remove-double-neg 60.9%

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

      7. associate-*l* 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in U* around inf 52.2%

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

      1. mul-1-neg 52.2%

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

      2. associate-/l* 53.0%

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

      3. distribute-neg-frac 53.0%

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

      4. *-commutative 53.0%

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

      5. associate-/r* 53.7%

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

      6. unpow2 53.7%

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

      7. unpow2 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in U around 0 52.2%

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

      1. associate-*r* 52.2%

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

      2. *-commutative 52.2%

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

      3. sub-neg 52.2%

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

      4. mul-1-neg 52.2%

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

      5. remove-double-neg 52.2%

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

      6. associate-*r* 52.2%

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

      7. unpow2 52.2%

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

      8. times-frac 58.2%

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

      9. unpow2 58.2%

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

   9. Simplified 58.2%

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

   if -1.12e-57 < n < -2.3499999999999999e-283

   1. Initial program 39.7%

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

      1. associate-*l* 55.2%

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

      2. sub-neg 55.2%

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

      3. associate-+l- 55.2%

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

      4. sub-neg 55.2%

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

      5. associate-/l* 60.2%

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

      6. remove-double-neg 60.2%

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

      7. associate-*l* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in Om around inf 53.6%

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

      1. unpow2 53.6%

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

      2. associate-*r/ 58.6%

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

   6. Simplified 58.6%

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

   if -2.3499999999999999e-283 < n < 7.0000000000000002e-59

   1. Initial program 46.6%

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

   2. Taylor expanded in Om around inf 43.5%

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

      1. unpow2 43.5%

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

      2. associate-*r/ 50.1%

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

   4. Simplified 50.1%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 8: 50.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;Om \leq -8 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= Om -8e-172)
     (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 t_1))))
     (if (<= Om 4.2e-215)
       (sqrt (* -2.0 (* (/ n Om) (* U (* l (* l (- 2.0 (* U* (/ n Om)))))))))
       (sqrt (* (* 2.0 n) (* U (- t (* 2.0 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 * (l / Om);
	double tmp;
	if (Om <= -8e-172) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else if (Om <= 4.2e-215) {
		tmp = sqrt((-2.0 * ((n / Om) * (U * (l * (l * (2.0 - (U_42_ * (n / Om)))))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * 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 * (l / om)
    if (om <= (-8d-172)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * t_1))))
    else if (om <= 4.2d-215) then
        tmp = sqrt(((-2.0d0) * ((n / om) * (u * (l * (l * (2.0d0 - (u_42 * (n / om)))))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * 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 * (l / Om);
	double tmp;
	if (Om <= -8e-172) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else if (Om <= 4.2e-215) {
		tmp = Math.sqrt((-2.0 * ((n / Om) * (U * (l * (l * (2.0 - (U_42_ * (n / Om)))))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if Om <= -8e-172:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))))
	elif Om <= 4.2e-215:
		tmp = math.sqrt((-2.0 * ((n / Om) * (U * (l * (l * (2.0 - (U_42_ * (n / Om)))))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (Om <= -8e-172)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * t_1))));
	elseif (Om <= 4.2e-215)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n / Om) * Float64(U * Float64(l * Float64(l * Float64(2.0 - Float64(U_42_ * Float64(n / Om)))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (Om <= -8e-172)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	elseif (Om <= 4.2e-215)
		tmp = sqrt((-2.0 * ((n / Om) * (U * (l * (l * (2.0 - (U_42_ * (n / Om)))))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * 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[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -8e-172], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 4.2e-215], N[Sqrt[N[(-2.0 * N[(N[(n / Om), $MachinePrecision] * N[(U * N[(l * N[(l * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -8.0000000000000003e-172

   1. Initial program 49.0%

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

   2. Taylor expanded in Om around inf 45.9%

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

      1. unpow2 45.9%

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

      2. associate-*r/ 54.8%

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

   4. Simplified 54.8%

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

   if -8.0000000000000003e-172 < Om < 4.2e-215

   1. Initial program 49.2%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in l around -inf 58.4%

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

      1. associate-/l* 52.0%

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

      2. associate-/r* 52.0%

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

      3. unpow2 52.0%

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

      4. *-commutative 52.0%

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

      5. mul-1-neg 52.0%

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

      6. unsub-neg 52.0%

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

      7. associate-/l* 49.4%

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

   6. Simplified 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. associate-/r/ 49.4%

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

      3. associate-/r/ 52.4%

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

   8. Applied egg-rr 52.4%

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

      1. *-lft-identity 52.4%

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

      2. associate-*r* 52.4%

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

      3. unpow2 52.4%

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

      4. associate-/r/ 57.0%

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

      5. unpow2 57.0%

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

      6. associate-*l/ 57.0%

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

      7. associate-*l/ 57.0%

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

      8. *-commutative 57.0%

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

   10. Simplified 57.0%

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

   11. Taylor expanded in U around 0 58.4%

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

       1. associate-/l* 52.1%

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

       2. associate-/r/ 58.9%

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

       3. associate-*r* 56.6%

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

       4. *-commutative 56.6%

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

       5. unpow2 56.6%

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

       6. associate-*l* 57.0%

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

       7. associate-/l* 52.2%

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

       8. associate-/r/ 57.0%

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

   13. Simplified 57.0%

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

   if 4.2e-215 < Om

   1. Initial program 50.2%

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

      1. associate-*l* 57.7%

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

      2. sub-neg 57.7%

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

      3. associate-+l- 57.7%

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

      4. sub-neg 57.7%

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

      5. associate-/l* 60.3%

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

      6. remove-double-neg 60.3%

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

      7. associate-*l* 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in Om around inf 49.9%

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

      1. unpow2 49.9%

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

      2. associate-*r/ 51.6%

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

   6. Simplified 51.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 9: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;Om \leq -1.45 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= Om -1.45e-173)
     (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 t_1))))
     (if (<= Om 1.15e-87)
       (sqrt (* (* -2.0 (* (* l l) (/ n Om))) (* U (- 2.0 (/ (* n U*) Om)))))
       (sqrt (* (* 2.0 n) (* U (- t (* 2.0 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 * (l / Om);
	double tmp;
	if (Om <= -1.45e-173) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else if (Om <= 1.15e-87) {
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * 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 * (l / om)
    if (om <= (-1.45d-173)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * t_1))))
    else if (om <= 1.15d-87) then
        tmp = sqrt((((-2.0d0) * ((l * l) * (n / om))) * (u * (2.0d0 - ((n * u_42) / om)))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * 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 * (l / Om);
	double tmp;
	if (Om <= -1.45e-173) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	} else if (Om <= 1.15e-87) {
		tmp = Math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if Om <= -1.45e-173:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))))
	elif Om <= 1.15e-87:
		tmp = math.sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (Om <= -1.45e-173)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * t_1))));
	elseif (Om <= 1.15e-87)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l * l) * Float64(n / Om))) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (Om <= -1.45e-173)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	elseif (Om <= 1.15e-87)
		tmp = sqrt(((-2.0 * ((l * l) * (n / Om))) * (U * (2.0 - ((n * U_42_) / Om)))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * 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[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.45e-173], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.15e-87], N[Sqrt[N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.4499999999999999e-173

   1. Initial program 49.0%

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

   2. Taylor expanded in Om around inf 45.9%

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

      1. unpow2 45.9%

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

      2. associate-*r/ 54.8%

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

   4. Simplified 54.8%

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

   if -1.4499999999999999e-173 < Om < 1.1500000000000001e-87

   1. Initial program 47.9%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in l around -inf 52.1%

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

      1. associate-/l* 47.9%

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

      2. associate-/r* 46.4%

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

      3. unpow2 46.4%

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

      4. *-commutative 46.4%

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

      5. mul-1-neg 46.4%

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

      6. unsub-neg 46.4%

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

      7. associate-/l* 43.0%

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

   6. Simplified 43.0%

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

      1. *-un-lft-identity 43.0%

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

      2. associate-/r/ 43.0%

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

      3. associate-/r/ 46.7%

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

   8. Applied egg-rr 46.7%

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

      1. *-lft-identity 46.7%

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

      2. associate-*r* 46.7%

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

      3. unpow2 46.7%

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

      4. associate-/r/ 51.2%

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

      5. unpow2 51.2%

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

      6. associate-*l/ 51.3%

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

      7. associate-*l/ 51.2%

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

      8. *-commutative 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in U* around inf 51.3%

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

   if 1.1500000000000001e-87 < Om

   1. Initial program 51.3%

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

      1. associate-*l* 58.2%

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

      2. sub-neg 58.2%

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

      3. associate-+l- 58.2%

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

      4. sub-neg 58.2%

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

      5. associate-/l* 61.4%

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

      6. remove-double-neg 61.4%

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

      7. associate-*l* 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in Om around inf 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*r/ 55.3%

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

   6. Simplified 55.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.45 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot \ell}}{\frac{-n}{\frac{Om}{U \cdot U*}}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 2.1e+143)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt (* -2.0 (/ n (/ (/ Om (* l l)) (/ (- n) (/ Om (* U U*)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.1e+143) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * (n / ((Om / (l * l)) / (-n / (Om / (U * U_42_)))))));
	}
	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 <= 2.1d+143) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * (n / ((om / (l * l)) / (-n / (om / (u * u_42)))))))
    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_ <= 2.1e+143) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (n / ((Om / (l * l)) / (-n / (Om / (U * U_42_)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 2.1e+143:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * (n / ((Om / (l * l)) / (-n / (Om / (U * U_42_)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 2.1e+143)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(n / Float64(Float64(Om / Float64(l * l)) / Float64(Float64(-n) / Float64(Om / Float64(U * U_42_)))))));
	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_ <= 2.1e+143)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * (n / ((Om / (l * l)) / (-n / (Om / (U * U_42_)))))));
	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$42$, 2.1e+143], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(n / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] / N[((-n) / N[(Om / N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 2.09999999999999988e143

   1. Initial program 48.4%

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

      1. associate-*l* 54.1%

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

      2. sub-neg 54.1%

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

      3. associate-+l- 54.1%

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

      4. sub-neg 54.1%

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

      5. associate-/l* 59.0%

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

      6. remove-double-neg 59.0%

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

      7. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in Om around inf 47.9%

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

      1. unpow2 47.9%

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

      2. associate-*r/ 52.3%

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

   6. Simplified 52.3%

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

   if 2.09999999999999988e143 < U*

   1. Initial program 54.4%

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

   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 l around -inf 41.5%

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

      1. associate-/l* 41.5%

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

      2. associate-/r* 42.8%

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

      3. unpow2 42.8%

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

      4. *-commutative 42.8%

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

      5. mul-1-neg 42.8%

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

      6. unsub-neg 42.8%

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

      7. associate-/l* 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in U* around inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. associate-/l* 44.5%

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

      3. distribute-neg-frac 44.5%

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

   9. Simplified 44.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 2.1 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot \ell}}{\frac{-n}{\frac{Om}{U \cdot U*}}}}}\\ \end{array} \]

Alternative 11: 47.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.12 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 1.12e+48)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (pow (* 2.0 (* t (* n U))) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.12e+48) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow((2.0 * (t * (n * U))), 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 (u <= 1.12d+48) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = (2.0d0 * (t * (n * u))) ** 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 (U <= 1.12e+48) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 1.12e+48:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 1.12e+48)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 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 (U <= 1.12e+48)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = (2.0 * (t * (n * U))) ^ 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[U, 1.12e+48], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.11999999999999995e48

   1. Initial program 47.0%

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

      1. associate-*l* 54.2%

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

      2. sub-neg 54.2%

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

      3. associate-+l- 54.2%

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

      4. sub-neg 54.2%

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

      5. associate-/l* 59.4%

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

      6. remove-double-neg 59.4%

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

      7. associate-*l* 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in Om around inf 45.0%

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

      1. unpow2 45.0%

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

      2. associate-*r/ 49.2%

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

   6. Simplified 49.2%

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

   if 1.11999999999999995e48 < U

   1. Initial program 63.0%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in t around inf 31.2%

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

      1. pow1/2 33.9%

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

      2. associate-*l* 33.9%

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

      3. *-commutative 33.9%

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

      4. associate-*r* 54.6%

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

   6. Applied egg-rr 54.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.12 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 12: 48.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;U \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om))))
   (if (<= U 3.5e-143)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1)))))
     (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 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 * (l / Om);
	double tmp;
	if (U <= 3.5e-143) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * 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 * (l / om)
    if (u <= 3.5d-143) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * 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 * (l / Om);
	double tmp;
	if (U <= 3.5e-143) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	tmp = 0
	if U <= 3.5e-143:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * t_1))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (U <= 3.5e-143)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * t_1))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	tmp = 0.0;
	if (U <= 3.5e-143)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	else
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * 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[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, 3.5e-143], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 3.50000000000000005e-143

   1. Initial program 45.8%

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

      1. associate-*l* 55.2%

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

      2. sub-neg 55.2%

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

      3. associate-+l- 55.2%

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

      4. sub-neg 55.2%

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

      5. associate-/l* 59.0%

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

      6. remove-double-neg 59.0%

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

      7. associate-*l* 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in Om around inf 46.1%

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

      1. unpow2 46.1%

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

      2. associate-*r/ 49.2%

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

   6. Simplified 49.2%

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

   if 3.50000000000000005e-143 < U

   1. Initial program 56.5%

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

   2. Taylor expanded in Om around inf 48.1%

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

      1. unpow2 48.1%

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

      2. associate-*r/ 54.4%

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

   4. Simplified 54.4%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 13: 40.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{+24}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.25e+24)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* -4.0 (* U (* (* l l) (/ n Om)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.25e+24) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((-4.0 * (U * ((l * l) * (n / 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 <= 2.25d+24) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * (u * ((l * l) * (n / 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 <= 2.25e+24) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * (U * ((l * l) * (n / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.25e+24:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * (U * ((l * l) * (n / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.25e+24)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64(Float64(l * l) * Float64(n / 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 <= 2.25e+24)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * (U * ((l * l) * (n / 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, 2.25e+24], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2500000000000001e24

   1. Initial program 53.5%

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

   2. Taylor expanded in t around inf 33.9%

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

      1. pow1/2 36.3%

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

      2. associate-*l* 39.8%

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

      3. associate-*r* 39.8%

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

   4. Applied egg-rr 39.8%

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

   if 2.2500000000000001e24 < l

   1. Initial program 31.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in l around -inf 39.2%

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

      1. associate-/l* 32.9%

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

      2. associate-/r* 32.9%

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

      3. unpow2 32.9%

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

      4. *-commutative 32.9%

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

      5. mul-1-neg 32.9%

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

      6. unsub-neg 32.9%

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

      7. associate-/l* 33.1%

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

   6. Simplified 33.1%

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

      1. *-un-lft-identity 33.1%

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

      2. associate-/r/ 35.1%

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

      3. associate-/r/ 35.1%

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

   8. Applied egg-rr 35.1%

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

      1. *-lft-identity 35.1%

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

      2. associate-*r* 35.2%

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

      3. unpow2 35.2%

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

      4. associate-/r/ 37.8%

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

      5. unpow2 37.8%

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

      6. associate-*l/ 37.8%

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

      7. associate-*l/ 37.8%

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

      8. *-commutative 37.8%

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

   10. Simplified 37.8%

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

   11. Taylor expanded in n around 0 22.7%

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

       1. associate-*r* 22.7%

          \[\leadsto \sqrt{-4 \cdot \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot U}}{Om}} \]

       2. associate-*l/ 22.8%

          \[\leadsto \sqrt{-4 \cdot \color{blue}{\left(\frac{n \cdot {\ell}^{2}}{Om} \cdot U\right)}} \]

       3. associate-*l/ 21.4%

          \[\leadsto \sqrt{-4 \cdot \left(\color{blue}{\left(\frac{n}{Om} \cdot {\ell}^{2}\right)} \cdot U\right)} \]

       4. unpow2 21.4%

          \[\leadsto \sqrt{-4 \cdot \left(\left(\frac{n}{Om} \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U\right)} \]

       5. *-commutative 21.4%

          \[\leadsto \sqrt{-4 \cdot \color{blue}{\left(U \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \ell\right)\right)\right)}} \]

       6. unpow2 21.4%

          \[\leadsto \sqrt{-4 \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \color{blue}{{\ell}^{2}}\right)\right)} \]

       7. *-commutative 21.4%

          \[\leadsto \sqrt{-4 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{n}{Om}\right)}\right)} \]

       8. unpow2 21.4%

          \[\leadsto \sqrt{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n}{Om}\right)\right)} \]

   13. Simplified 21.4%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{+24}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]

Alternative 14: 39.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.2e+21)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (/ (* -4.0 (* n (* U (* 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 (l <= 5.2e+21) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt(((-4.0 * (n * (U * (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 (l <= 5.2d+21) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((((-4.0d0) * (n * (u * (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 (l <= 5.2e+21) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.2e+21:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt(((-4.0 * (n * (U * (l * l)))) / Om))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.2e+21)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l * 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 (l <= 5.2e+21)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt(((-4.0 * (n * (U * (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[l, 5.2e+21], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.2e21

   1. Initial program 53.5%

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

   2. Taylor expanded in t around inf 33.9%

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

      1. pow1/2 36.3%

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

      2. associate-*l* 39.8%

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

      3. associate-*r* 39.8%

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

   4. Applied egg-rr 39.8%

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

   if 5.2e21 < l

   1. Initial program 31.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in l around -inf 39.2%

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

      1. associate-/l* 32.9%

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

      2. associate-/r* 32.9%

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

      3. unpow2 32.9%

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

      4. *-commutative 32.9%

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

      5. mul-1-neg 32.9%

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

      6. unsub-neg 32.9%

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

      7. associate-/l* 33.1%

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

   6. Simplified 33.1%

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

      1. *-un-lft-identity 33.1%

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

      2. associate-/r/ 35.1%

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

      3. associate-/r/ 35.1%

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

   8. Applied egg-rr 35.1%

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

      1. *-lft-identity 35.1%

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

      2. associate-*r* 35.2%

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

      3. unpow2 35.2%

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

      4. associate-/r/ 37.8%

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

      5. unpow2 37.8%

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

      6. associate-*l/ 37.8%

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

      7. associate-*l/ 37.8%

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

      8. *-commutative 37.8%

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

   10. Simplified 37.8%

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

   11. Taylor expanded in n around 0 22.7%

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

       1. associate-*r/ 23.0%

          \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)}{Om}}} \]

       2. *-commutative 23.0%

          \[\leadsto \sqrt{\frac{-4 \cdot \left(n \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}\right)}{Om}} \]

       3. unpow2 23.0%

          \[\leadsto \sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{Om}} \]

   13. Simplified 23.0%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 15: 38.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 1.5e+49)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* (* (* 2.0 n) U) t))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.5e+49) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((((2.0 * n) * U) * t));
	}
	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 <= 1.5d+49) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((((2.0d0 * n) * u) * t))
    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 <= 1.5e+49) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 1.5e+49:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 1.5e+49)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.5e+49)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((((2.0 * n) * U) * t));
	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, 1.5e+49], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.5000000000000001e49

   1. Initial program 47.2%

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

   2. Taylor expanded in t around inf 27.8%

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

      1. pow1/2 29.2%

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

      2. associate-*l* 35.9%

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

      3. associate-*r* 35.9%

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

   4. Applied egg-rr 35.9%

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

   if 1.5000000000000001e49 < U

   1. Initial program 62.1%

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

   2. Taylor expanded in t around inf 48.3%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \]

Alternative 16: 38.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 4.6 \cdot 10^{-144}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 4.6e-144)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (pow (* 2.0 (* t (* n U))) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 4.6e-144) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = pow((2.0 * (t * (n * U))), 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 (u <= 4.6d-144) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (t * (n * u))) ** 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 (U <= 4.6e-144) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 4.6e-144:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 4.6e-144)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 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 (U <= 4.6e-144)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = (2.0 * (t * (n * U))) ^ 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[U, 4.6e-144], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 4.6e-144

   1. Initial program 45.8%

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

   2. Taylor expanded in t around inf 27.2%

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

      1. pow1/2 28.4%

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

      2. associate-*l* 37.1%

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

      3. associate-*r* 37.1%

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

   4. Applied egg-rr 37.1%

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

   if 4.6e-144 < U

   1. Initial program 56.5%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in t around inf 30.7%

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

      1. pow1/2 32.0%

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

      2. associate-*l* 32.0%

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

      3. *-commutative 32.0%

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

      4. associate-*r* 41.6%

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

   6. Applied egg-rr 41.6%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.6 \cdot 10^{-144}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 17: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 3.5e-143)
   (sqrt (* (* 2.0 n) (* U t)))
   (sqrt (* (* (* 2.0 n) U) t))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 3.5e-143) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t));
	}
	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 <= 3.5d-143) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t))
    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 <= 3.5e-143) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 3.5e-143:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 3.5e-143)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 3.5e-143)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt((((2.0 * n) * U) * t));
	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, 3.5e-143], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 3.50000000000000005e-143

   1. Initial program 45.8%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in t around inf 35.8%

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

   if 3.50000000000000005e-143 < U

   1. Initial program 56.5%

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

   2. Taylor expanded in t around inf 38.2%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \]

Alternative 18: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.6%

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

3. Simplified 59.0%

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

4. Taylor expanded in t around inf 34.0%

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

5. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2023242 
(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*))))))