Toniolo and Linder, Equation (13)

Percentage Accurate: 49.4% → 68.0%

Time: 20.0s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_1 0.0)
     (sqrt (* 2.0 (* U (* n (fma -2.0 (* l (/ l Om)) t)))))
     (if (<= t_1 4e+296)
       (sqrt t_1)
       (*
        (* l (sqrt 2.0))
        (sqrt (/ (* n (* U (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double 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 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * fma(-2.0, (l * (l / Om)), t)))));
	} else if (t_1 <= 4e+296) {
		tmp = sqrt(t_1);
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 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 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(-2.0, Float64(l * Float64(l / Om)), t)))));
	elseif (t_1 <= 4e+296)
		tmp = sqrt(t_1);
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 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, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 4e+296], N[Sqrt[t$95$1], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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*)))) < 0.0

   1. Initial program 13.1%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in n around 0 40.5%

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

      1. associate-*r* 40.5%

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

      2. +-commutative 40.5%

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

      3. unpow2 40.5%

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

      4. fma-def 40.5%

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

      5. associate-*r/ 43.1%

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

   6. Simplified 43.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 3.99999999999999993e296

   1. Initial program 95.3%

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

   if 3.99999999999999993e296 < (*.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 23.7%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around inf 44.8%

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

   5. Taylor expanded in l around inf 29.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 2: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{2}{Om}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.5e+124)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
   (*
    (sqrt 2.0)
    (* l (sqrt (* n (* U (- (/ n (/ (* Om Om) U*)) (/ 2.0 Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.5e+124) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n * (U * ((n / ((Om * Om) / U_42_)) - (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) :: tmp
    if (l <= 2.5d+124) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt(2.0d0) * (l * sqrt((n * (u * ((n / ((om * om) / u_42)) - (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 tmp;
	if (l <= 2.5e+124) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((n * (U * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.5e+124:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((n * (U * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.5e+124)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n * Float64(U * Float64(Float64(n / Float64(Float64(Om * Om) / U_42_)) - Float64(2.0 / 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 <= 2.5e+124)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(2.0) * (l * sqrt((n * (U * ((n / ((Om * Om) / U_42_)) - (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$42$_] := If[LessEqual[l, 2.5e+124], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n * N[(U * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4999999999999998e124

   1. Initial program 53.9%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in U around 0 59.4%

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

   if 2.4999999999999998e124 < l

   1. Initial program 41.5%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in U around 0 49.1%

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

   5. Taylor expanded in l around inf 74.6%

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

      1. associate-*l* 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-/l* 77.7%

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

      4. unpow2 77.7%

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

      5. associate-*r/ 77.7%

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

      6. metadata-eval 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 3: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.3e+123)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
   (*
    (* l (sqrt 2.0))
    (sqrt (/ (* n (* U (- (/ (* n (- U* U)) Om) 2.0))) Om)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.3e+123) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * (U_42_ - U)) / 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) :: tmp
    if (l <= 1.3d+123) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((n * (u * (((n * (u_42 - u)) / 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 tmp;
	if (l <= 1.3e+123) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.3e+123:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.3e+123)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / 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 <= 1.3e+123)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * (U_42_ - U)) / 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$42$_] := If[LessEqual[l, 1.3e+123], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.29999999999999993e123

   1. Initial program 54.1%

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

   3. Simplified 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 U around 0 59.7%

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

   if 1.29999999999999993e123 < l

   1. Initial program 40.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in t around inf 55.6%

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

   5. Taylor expanded in l around inf 78.6%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 4: 57.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\\ t_2 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ t_3 := n \cdot \left(U \cdot \ell\right)\\ \mathbf{if}\;Om \leq -1.5 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{t_1 - 2 \cdot \left(\left(\ell \cdot -2 - \frac{\left(U - U*\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(t_3 \cdot \frac{-1}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.75 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 - \left(U* \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{t_1 + 2 \cdot \frac{t_3 \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n (* U t))))
        (t_2 (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om))))))))
        (t_3 (* n (* U l))))
   (if (<= Om -1.5e+149)
     t_2
     (if (<= Om -1.7e-244)
       (sqrt
        (-
         t_1
         (*
          2.0
          (* (- (* l -2.0) (/ (* (- U U*) (* n l)) Om)) (* t_3 (/ -1.0 Om))))))
       (if (<= Om 2.75e-149)
         (sqrt
          (*
           (* 2.0 n)
           (*
            U
            (+ t (/ (* l (- (* l -2.0) (* (* U* (* n l)) (/ -1.0 Om)))) Om)))))
         (if (<= Om 8.5e+165)
           (sqrt
            (+
             t_1
             (*
              2.0
              (/ (* t_3 (+ (* l -2.0) (/ (* n (* l (- U* U))) Om))) Om))))
           t_2))))))

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));
	double t_2 = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	double t_3 = n * (U * l);
	double tmp;
	if (Om <= -1.5e+149) {
		tmp = t_2;
	} else if (Om <= -1.7e-244) {
		tmp = sqrt((t_1 - (2.0 * (((l * -2.0) - (((U - U_42_) * (n * l)) / Om)) * (t_3 * (-1.0 / Om))))));
	} else if (Om <= 2.75e-149) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	} else if (Om <= 8.5e+165) {
		tmp = sqrt((t_1 + (2.0 * ((t_3 * ((l * -2.0) + ((n * (l * (U_42_ - U))) / Om))) / Om))));
	} 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) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 * (n * (u * t))
    t_2 = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    t_3 = n * (u * l)
    if (om <= (-1.5d+149)) then
        tmp = t_2
    else if (om <= (-1.7d-244)) then
        tmp = sqrt((t_1 - (2.0d0 * (((l * (-2.0d0)) - (((u - u_42) * (n * l)) / om)) * (t_3 * ((-1.0d0) / om))))))
    else if (om <= 2.75d-149) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) - ((u_42 * (n * l)) * ((-1.0d0) / om)))) / om)))))
    else if (om <= 8.5d+165) then
        tmp = sqrt((t_1 + (2.0d0 * ((t_3 * ((l * (-2.0d0)) + ((n * (l * (u_42 - u))) / om))) / om))))
    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 = 2.0 * (n * (U * t));
	double t_2 = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	double t_3 = n * (U * l);
	double tmp;
	if (Om <= -1.5e+149) {
		tmp = t_2;
	} else if (Om <= -1.7e-244) {
		tmp = Math.sqrt((t_1 - (2.0 * (((l * -2.0) - (((U - U_42_) * (n * l)) / Om)) * (t_3 * (-1.0 / Om))))));
	} else if (Om <= 2.75e-149) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	} else if (Om <= 8.5e+165) {
		tmp = Math.sqrt((t_1 + (2.0 * ((t_3 * ((l * -2.0) + ((n * (l * (U_42_ - U))) / Om))) / Om))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * (U * t))
	t_2 = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	t_3 = n * (U * l)
	tmp = 0
	if Om <= -1.5e+149:
		tmp = t_2
	elif Om <= -1.7e-244:
		tmp = math.sqrt((t_1 - (2.0 * (((l * -2.0) - (((U - U_42_) * (n * l)) / Om)) * (t_3 * (-1.0 / Om))))))
	elif Om <= 2.75e-149:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))))
	elif Om <= 8.5e+165:
		tmp = math.sqrt((t_1 + (2.0 * ((t_3 * ((l * -2.0) + ((n * (l * (U_42_ - U))) / Om))) / Om))))
	else:
		tmp = t_2
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * Float64(U * t)))
	t_2 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))))
	t_3 = Float64(n * Float64(U * l))
	tmp = 0.0
	if (Om <= -1.5e+149)
		tmp = t_2;
	elseif (Om <= -1.7e-244)
		tmp = sqrt(Float64(t_1 - Float64(2.0 * Float64(Float64(Float64(l * -2.0) - Float64(Float64(Float64(U - U_42_) * Float64(n * l)) / Om)) * Float64(t_3 * Float64(-1.0 / Om))))));
	elseif (Om <= 2.75e-149)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) - Float64(Float64(U_42_ * Float64(n * l)) * Float64(-1.0 / Om)))) / Om)))));
	elseif (Om <= 8.5e+165)
		tmp = sqrt(Float64(t_1 + Float64(2.0 * Float64(Float64(t_3 * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * Float64(U_42_ - U))) / Om))) / Om))));
	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 = 2.0 * (n * (U * t));
	t_2 = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	t_3 = n * (U * l);
	tmp = 0.0;
	if (Om <= -1.5e+149)
		tmp = t_2;
	elseif (Om <= -1.7e-244)
		tmp = sqrt((t_1 - (2.0 * (((l * -2.0) - (((U - U_42_) * (n * l)) / Om)) * (t_3 * (-1.0 / Om))))));
	elseif (Om <= 2.75e-149)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	elseif (Om <= 8.5e+165)
		tmp = sqrt((t_1 + (2.0 * ((t_3 * ((l * -2.0) + ((n * (l * (U_42_ - U))) / Om))) / Om))));
	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[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.5e+149], t$95$2, If[LessEqual[Om, -1.7e-244], N[Sqrt[N[(t$95$1 - N[(2.0 * N[(N[(N[(l * -2.0), $MachinePrecision] - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.75e-149], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 8.5e+165], N[Sqrt[N[(t$95$1 + N[(2.0 * N[(N[(t$95$3 * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -1.50000000000000002e149 or 8.5000000000000001e165 < Om

   1. Initial program 53.9%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in n around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. unpow2 57.6%

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

      3. associate-*r/ 66.4%

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

   6. Simplified 66.4%

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

   if -1.50000000000000002e149 < Om < -1.70000000000000004e-244

   1. Initial program 52.9%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in t around inf 61.6%

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

      1. div-inv 61.6%

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

      2. associate-/l* 59.1%

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

      3. *-commutative 59.1%

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

   6. Applied egg-rr 59.1%

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

      1. associate-*l* 58.2%

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

      2. associate-/l* 60.9%

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

      3. +-commutative 60.9%

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

      4. *-commutative 60.9%

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

      5. associate-*l/ 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in n around 0 60.9%

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

       1. associate-*r* 66.4%

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

   11. Simplified 66.4%

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

   if -1.70000000000000004e-244 < Om < 2.75000000000000021e-149

   1. Initial program 51.6%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in U around 0 74.7%

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

      1. div-inv 74.6%

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

      2. associate-*r* 74.8%

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

   6. Applied egg-rr 74.8%

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

   if 2.75000000000000021e-149 < Om < 8.5000000000000001e165

   1. Initial program 49.5%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in t around inf 66.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.5 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.7 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \left(\left(\ell \cdot -2 - \frac{\left(U - U*\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.75 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 - \left(U* \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot \left(U* - U\right)\right)}{Om}\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 5: 57.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 - \left(U* \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \left(\left(\ell \cdot -2 + \left(\ell \cdot \left(U* - U\right)\right) \cdot \frac{n}{Om}\right) \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.2e+71)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (- (* l -2.0) (* (* U* (* n l)) (/ -1.0 Om)))) Om)))))
   (sqrt
    (-
     (* 2.0 (* n (* U t)))
     (*
      2.0
      (*
       (+ (* l -2.0) (* (* l (- U* U)) (/ n Om)))
       (* (* n (* U l)) (/ -1.0 Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+71) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	} else {
		tmp = sqrt(((2.0 * (n * (U * t))) - (2.0 * (((l * -2.0) + ((l * (U_42_ - U)) * (n / Om))) * ((n * (U * l)) * (-1.0 / 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) :: tmp
    if (l <= 6.2d+71) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) - ((u_42 * (n * l)) * ((-1.0d0) / om)))) / om)))))
    else
        tmp = sqrt(((2.0d0 * (n * (u * t))) - (2.0d0 * (((l * (-2.0d0)) + ((l * (u_42 - u)) * (n / om))) * ((n * (u * l)) * ((-1.0d0) / 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 tmp;
	if (l <= 6.2e+71) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * (U * t))) - (2.0 * (((l * -2.0) + ((l * (U_42_ - U)) * (n / Om))) * ((n * (U * l)) * (-1.0 / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.2e+71:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))))
	else:
		tmp = math.sqrt(((2.0 * (n * (U * t))) - (2.0 * (((l * -2.0) + ((l * (U_42_ - U)) * (n / Om))) * ((n * (U * l)) * (-1.0 / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.2e+71)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) - Float64(Float64(U_42_ * Float64(n * l)) * Float64(-1.0 / Om)))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) - Float64(2.0 * Float64(Float64(Float64(l * -2.0) + Float64(Float64(l * Float64(U_42_ - U)) * Float64(n / Om))) * Float64(Float64(n * Float64(U * l)) * Float64(-1.0 / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.2e+71)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) - ((U_42_ * (n * l)) * (-1.0 / Om)))) / Om)))));
	else
		tmp = sqrt(((2.0 * (n * (U * t))) - (2.0 * (((l * -2.0) + ((l * (U_42_ - U)) * (n / Om))) * ((n * (U * l)) * (-1.0 / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.2e+71], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.20000000000000036e71

   1. Initial program 55.8%

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

   3. Simplified 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 U around 0 61.6%

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

      1. div-inv 61.6%

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

      2. associate-*r* 61.1%

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

   6. Applied egg-rr 61.1%

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

   if 6.20000000000000036e71 < l

   1. Initial program 35.4%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in t around inf 47.2%

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

      1. div-inv 47.2%

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

      2. associate-/l* 49.0%

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

      3. *-commutative 49.0%

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

   6. Applied egg-rr 49.0%

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

      1. associate-*l* 49.7%

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

      2. associate-/l* 47.9%

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

      3. +-commutative 47.9%

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

      4. *-commutative 47.9%

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

      5. associate-*l/ 50.0%

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

   8. Simplified 50.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 - \left(U* \cdot \left(n \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \left(\left(\ell \cdot -2 + \left(\ell \cdot \left(U* - U\right)\right) \cdot \frac{n}{Om}\right) \cdot \left(\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-1}{Om}\right)\right)}\\ \end{array} \]

Alternative 6: 50.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.4 \cdot 10^{-117} \lor \neg \left(Om \leq 1.45 \cdot 10^{-117}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -2.4e-117) (not (<= Om 1.45e-117)))
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om)))))))
   (sqrt
    (* -2.0 (* (/ n Om) (* l (* l (* U (+ 2.0 (/ (* n (- U U*)) Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.4e-117) || !(Om <= 1.45e-117)) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-2.4d-117)) .or. (.not. (om <= 1.45d-117))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * ((n / om) * (l * (l * (u * (2.0d0 + ((n * (u - u_42)) / om))))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.4e-117) || !(Om <= 1.45e-117)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -2.4e-117) or not (Om <= 1.45e-117):
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -2.4e-117) || !(Om <= 1.45e-117))
		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(Float64(n / Om) * Float64(l * Float64(l * Float64(U * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -2.4e-117) || ~((Om <= 1.45e-117)))
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -2.4e-117], N[Not[LessEqual[Om, 1.45e-117]], $MachinePrecision]], 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 / Om), $MachinePrecision] * N[(l * N[(l * N[(U * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.40000000000000014e-117 or 1.45e-117 < Om

   1. Initial program 52.1%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in n around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. associate-*r/ 59.4%

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

   6. Simplified 59.4%

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

   if -2.40000000000000014e-117 < Om < 1.45e-117

   1. Initial program 53.0%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in l around -inf 58.1%

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

      1. unpow2 58.1%

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

      2. *-commutative 58.1%

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

      3. mul-1-neg 58.1%

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

      4. associate-/l* 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in l around 0 54.6%

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

      1. associate-/l* 46.7%

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

      2. *-commutative 46.7%

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

      3. associate-*l/ 46.6%

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

      4. associate-*r* 50.1%

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

      5. unpow2 50.1%

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

      6. associate-/r/ 61.2%

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

      7. associate-*l* 64.9%

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

      8. associate-*l/ 64.9%

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

   9. Simplified 64.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.4 \cdot 10^{-117} \lor \neg \left(Om \leq 1.45 \cdot 10^{-117}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7: 56.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.2e+80)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
   (if (<= l 7.6e+208)
     (sqrt
      (* -2.0 (* (/ n Om) (* l (* l (* U (+ 2.0 (/ (* n (- U U*)) Om))))))))
     (pow (* -4.0 (/ (* U (* l (* n l))) Om)) 0.5))))

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+80) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 7.6e+208) {
		tmp = sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))));
	} else {
		tmp = pow((-4.0 * ((U * (l * (n * l))) / Om)), 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 (l <= 5.2d+80) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else if (l <= 7.6d+208) then
        tmp = sqrt(((-2.0d0) * ((n / om) * (l * (l * (u * (2.0d0 + ((n * (u - u_42)) / om))))))))
    else
        tmp = ((-4.0d0) * ((u * (l * (n * l))) / om)) ** 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 (l <= 5.2e+80) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else if (l <= 7.6e+208) {
		tmp = Math.sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))));
	} else {
		tmp = Math.pow((-4.0 * ((U * (l * (n * l))) / Om)), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.2e+80:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	elif l <= 7.6e+208:
		tmp = math.sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))))
	else:
		tmp = math.pow((-4.0 * ((U * (l * (n * l))) / Om)), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.2e+80)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	elseif (l <= 7.6e+208)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n / Om) * Float64(l * Float64(l * Float64(U * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))))))));
	else
		tmp = Float64(-4.0 * Float64(Float64(U * Float64(l * Float64(n * l))) / Om)) ^ 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 (l <= 5.2e+80)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	elseif (l <= 7.6e+208)
		tmp = sqrt((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + ((n * (U - U_42_)) / Om))))))));
	else
		tmp = (-4.0 * ((U * (l * (n * l))) / Om)) ^ 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[l, 5.2e+80], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.6e+208], N[Sqrt[N[(-2.0 * N[(N[(n / Om), $MachinePrecision] * N[(l * N[(l * N[(U * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(N[(U * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.19999999999999963e80

   1. Initial program 54.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in U around 0 60.6%

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

   if 5.19999999999999963e80 < l < 7.6000000000000004e208

   1. Initial program 46.2%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in l around -inf 60.6%

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

      1. unpow2 60.6%

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

      2. *-commutative 60.6%

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

      3. mul-1-neg 60.6%

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

      4. associate-/l* 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in l around 0 60.6%

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

      1. associate-/l* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*l/ 46.8%

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

      4. associate-*r* 46.8%

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

      5. unpow2 46.8%

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

      6. associate-/r/ 60.9%

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

      7. associate-*l* 69.5%

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

      8. associate-*l/ 69.5%

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

   9. Simplified 69.5%

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

   if 7.6000000000000004e208 < l

   1. Initial program 28.8%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in l around -inf 41.8%

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

      1. unpow2 41.8%

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

      2. *-commutative 41.8%

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

      3. mul-1-neg 41.8%

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

      4. associate-/l* 41.8%

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

   6. Simplified 41.8%

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

   7. Taylor expanded in n around 0 30.2%

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

      1. associate-*r* 30.2%

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

      2. unpow2 30.2%

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

   9. Simplified 30.2%

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

       1. pow1/2 41.8%

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

       2. associate-*r* 41.8%

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

       3. metadata-eval 41.8%

          \[\leadsto {\left(\color{blue}{-4} \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)}^{0.5} \]

       4. *-commutative 41.8%

          \[\leadsto {\left(-4 \cdot \frac{\color{blue}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}}{Om}\right)}^{0.5} \]

       5. associate-*r* 57.7%

          \[\leadsto {\left(-4 \cdot \frac{U \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)}}{Om}\right)}^{0.5} \]

   11. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)}{Om}\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 8: 49.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.35 \cdot 10^{-117} \lor \neg \left(Om \leq 1.6 \cdot 10^{-117}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.35e-117) (not (<= Om 1.6e-117)))
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om)))))))
   (sqrt (* -2.0 (/ (* n (* (* l l) (* U (- 2.0 (/ (* n U*) Om))))) Om)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.35e-117) || !(Om <= 1.6e-117)) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * ((n * ((l * l) * (U * (2.0 - ((n * U_42_) / Om))))) / Om)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.35d-117)) .or. (.not. (om <= 1.6d-117))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * ((n * ((l * l) * (u * (2.0d0 - ((n * u_42) / om))))) / om)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.35e-117) || !(Om <= 1.6e-117)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((n * ((l * l) * (U * (2.0 - ((n * U_42_) / Om))))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.35e-117) or not (Om <= 1.6e-117):
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * ((n * ((l * l) * (U * (2.0 - ((n * U_42_) / Om))))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.35e-117) || !(Om <= 1.6e-117))
		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(Float64(n * Float64(Float64(l * l) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om))))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.35e-117) || ~((Om <= 1.6e-117)))
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * ((n * ((l * l) * (U * (2.0 - ((n * U_42_) / Om))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.35e-117], N[Not[LessEqual[Om, 1.6e-117]], $MachinePrecision]], 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[(N[(l * l), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.35000000000000001e-117 or 1.59999999999999998e-117 < Om

   1. Initial program 52.1%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in n around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. associate-*r/ 59.4%

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

   6. Simplified 59.4%

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

   if -1.35000000000000001e-117 < Om < 1.59999999999999998e-117

   1. Initial program 53.0%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in l around -inf 58.1%

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

      1. unpow2 58.1%

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

      2. *-commutative 58.1%

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

      3. mul-1-neg 58.1%

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

      4. associate-/l* 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in U* around inf 57.0%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.35 \cdot 10^{-117} \lor \neg \left(Om \leq 1.6 \cdot 10^{-117}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 9: 47.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.85 \cdot 10^{-117} \lor \neg \left(Om \leq 3 \cdot 10^{-123}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \frac{-n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}}}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.85e-117) (not (<= Om 3e-123)))
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om)))))))
   (sqrt (* -2.0 (/ (* n (/ (- n) (/ Om (* (* l l) (* U U*))))) Om)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.85e-117) || !(Om <= 3e-123)) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * ((n * (-n / (Om / ((l * l) * (U * U_42_))))) / Om)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.85d-117)) .or. (.not. (om <= 3d-123))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * ((n * (-n / (om / ((l * l) * (u * u_42))))) / om)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.85e-117) || !(Om <= 3e-123)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((n * (-n / (Om / ((l * l) * (U * U_42_))))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.85e-117) or not (Om <= 3e-123):
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * ((n * (-n / (Om / ((l * l) * (U * U_42_))))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.85e-117) || !(Om <= 3e-123))
		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(Float64(n * Float64(Float64(-n) / Float64(Om / Float64(Float64(l * l) * Float64(U * U_42_))))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.85e-117) || ~((Om <= 3e-123)))
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * ((n * (-n / (Om / ((l * l) * (U * U_42_))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.85e-117], N[Not[LessEqual[Om, 3e-123]], $MachinePrecision]], 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[((-n) / N[(Om / N[(N[(l * l), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.8500000000000001e-117 or 2.99999999999999984e-123 < Om

   1. Initial program 51.9%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in n around 0 54.2%

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

      1. *-commutative 54.2%

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

      2. unpow2 54.2%

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

      3. associate-*r/ 59.1%

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

   6. Simplified 59.1%

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

   if -1.8500000000000001e-117 < Om < 2.99999999999999984e-123

   1. Initial program 53.9%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in l around -inf 57.3%

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

      1. unpow2 57.3%

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

      2. *-commutative 57.3%

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

      3. mul-1-neg 57.3%

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

      4. associate-/l* 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in U* around inf 49.2%

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

      1. mul-1-neg 49.2%

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

      2. associate-/l* 49.3%

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

      3. distribute-neg-frac 49.3%

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

      4. *-commutative 49.3%

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

      5. unpow2 49.3%

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

   9. Simplified 49.3%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.85 \cdot 10^{-117} \lor \neg \left(Om \leq 3 \cdot 10^{-123}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot \frac{-n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}}}{Om}}\\ \end{array} \]

Alternative 10: 49.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.8e+165)
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om)))))))
   (pow (* -4.0 (/ (* U (* l (* n l))) Om)) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.8e+165) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow((-4.0 * ((U * (l * (n * l))) / Om)), 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 (l <= 5.8d+165) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    else
        tmp = ((-4.0d0) * ((u * (l * (n * l))) / om)) ** 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 (l <= 5.8e+165) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow((-4.0 * ((U * (l * (n * l))) / Om)), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.8e+165:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow((-4.0 * ((U * (l * (n * l))) / Om)), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.8e+165)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(-4.0 * Float64(Float64(U * Float64(l * Float64(n * l))) / Om)) ^ 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 (l <= 5.8e+165)
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	else
		tmp = (-4.0 * ((U * (l * (n * l))) / Om)) ^ 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[l, 5.8e+165], 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[(-4.0 * N[(N[(U * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.80000000000000011e165

   1. Initial program 53.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in n around 0 50.3%

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

      1. *-commutative 50.3%

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

      2. unpow2 50.3%

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

      3. associate-*r/ 53.6%

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

   6. Simplified 53.6%

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

   if 5.80000000000000011e165 < l

   1. Initial program 38.9%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in l around -inf 50.1%

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

      1. unpow2 50.1%

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

      2. *-commutative 50.1%

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

      3. mul-1-neg 50.1%

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

      4. associate-/l* 50.1%

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

   6. Simplified 50.1%

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

   7. Taylor expanded in n around 0 40.2%

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

      1. associate-*r* 40.2%

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

      2. unpow2 40.2%

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

   9. Simplified 40.2%

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

       1. pow1/2 50.1%

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

       2. associate-*r* 50.1%

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

       3. metadata-eval 50.1%

          \[\leadsto {\left(\color{blue}{-4} \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)}^{0.5} \]

       4. *-commutative 50.1%

          \[\leadsto {\left(-4 \cdot \frac{\color{blue}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}}{Om}\right)}^{0.5} \]

       5. associate-*r* 63.8%

          \[\leadsto {\left(-4 \cdot \frac{U \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)}}{Om}\right)}^{0.5} \]

   11. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)}{Om}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 11: 47.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 9.2e+39)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (pow (* -4.0 (/ (* U (* l (* n l))) Om)) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 9.2e+39) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = pow((-4.0 * ((U * (l * (n * l))) / Om)), 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 (l <= 9.2d+39) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = ((-4.0d0) * ((u * (l * (n * l))) / om)) ** 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 (l <= 9.2e+39) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = Math.pow((-4.0 * ((U * (l * (n * l))) / Om)), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 9.2e+39:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = math.pow((-4.0 * ((U * (l * (n * l))) / Om)), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 9.2e+39)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = Float64(-4.0 * Float64(Float64(U * Float64(l * Float64(n * l))) / Om)) ^ 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 (l <= 9.2e+39)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = (-4.0 * ((U * (l * (n * l))) / Om)) ^ 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[l, 9.2e+39], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(-4.0 * N[(N[(U * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.20000000000000047e39

   1. Initial program 55.3%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in t around inf 43.8%

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

      1. pow1/2 44.4%

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

   6. Applied egg-rr 44.4%

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

   if 9.20000000000000047e39 < l

   1. Initial program 40.4%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in l around -inf 45.5%

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

      1. unpow2 45.5%

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

      2. *-commutative 45.5%

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

      3. mul-1-neg 45.5%

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

      4. associate-/l* 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in n around 0 27.6%

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

      1. associate-*r* 31.4%

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

      2. unpow2 31.4%

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

   9. Simplified 31.4%

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

       1. pow1/2 41.7%

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

       2. associate-*r* 41.7%

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

       3. metadata-eval 41.7%

          \[\leadsto {\left(\color{blue}{-4} \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}\right)}^{0.5} \]

       4. *-commutative 41.7%

          \[\leadsto {\left(-4 \cdot \frac{\color{blue}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}}{Om}\right)}^{0.5} \]

       5. associate-*r* 47.4%

          \[\leadsto {\left(-4 \cdot \frac{U \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)}}{Om}\right)}^{0.5} \]

   11. Applied egg-rr 47.4%

       \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)}{Om}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 12: 39.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+40}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot \ell}}{U}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.1e+40)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (sqrt (* -4.0 (/ n (/ (/ Om (* l l)) U))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.1e+40) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = sqrt((-4.0 * (n / ((Om / (l * l)) / 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) :: tmp
    if (l <= 1.1d+40) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * (n / ((om / (l * l)) / 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 tmp;
	if (l <= 1.1e+40) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * (n / ((Om / (l * l)) / U))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.1e+40:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = math.sqrt((-4.0 * (n / ((Om / (l * l)) / U))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.1e+40)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(n / Float64(Float64(Om / Float64(l * l)) / U))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.1e+40)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = sqrt((-4.0 * (n / ((Om / (l * l)) / U))));
	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, 1.1e+40], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(n / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0999999999999999e40

   1. Initial program 55.3%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in t around inf 43.8%

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

      1. pow1/2 44.4%

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

   6. Applied egg-rr 44.4%

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

   if 1.0999999999999999e40 < l

   1. Initial program 40.4%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in l around -inf 45.5%

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

      1. unpow2 45.5%

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

      2. *-commutative 45.5%

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

      3. mul-1-neg 45.5%

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

      4. associate-/l* 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in n around 0 27.6%

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

      1. associate-*r* 31.4%

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

      2. unpow2 31.4%

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

   9. Simplified 31.4%

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

       1. *-un-lft-identity 31.4%

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

       2. associate-*r* 31.4%

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

       3. metadata-eval 31.4%

          \[\leadsto 1 \cdot \sqrt{\color{blue}{-4} \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}} \]

       4. *-commutative 31.4%

          \[\leadsto 1 \cdot \sqrt{-4 \cdot \frac{\color{blue}{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}}{Om}} \]

       5. associate-*r* 37.0%

          \[\leadsto 1 \cdot \sqrt{-4 \cdot \frac{U \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)}}{Om}} \]

   11. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{1 \cdot \sqrt{-4 \cdot \frac{U \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)}{Om}}} \]
   12. Step-by-step derivation

       1. *-lft-identity 37.0%

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

       2. *-commutative 37.0%

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

       3. associate-*l* 31.4%

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

       4. unpow2 31.4%

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

       5. associate-*r* 27.6%

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

       6. associate-/l* 27.1%

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

       7. associate-/r* 27.5%

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

       8. unpow2 27.5%

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

   13. Simplified 27.5%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+40}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{n}{\frac{\frac{Om}{\ell \cdot \ell}}{U}}}\\ \end{array} \]

Alternative 13: 36.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = ((2.0d0 * n) * (u * t)) ** 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

3. Simplified 59.4%

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

4. Taylor expanded in t around inf 37.8%

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

   1. pow1/2 39.5%

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

6. Applied egg-rr 39.5%

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

7. Final simplification 39.5%

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

Alternative 14: 35.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

3. Simplified 59.4%

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

4. Taylor expanded in t around inf 37.8%

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

   1. pow1/2 39.5%

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

6. Applied egg-rr 39.5%

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

   1. unpow-prod-down 22.8%

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

   2. *-commutative 22.8%

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

   3. *-commutative 22.8%

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

   4. pow1/2 22.4%

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

8. Applied egg-rr 22.4%

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

   1. unpow1/2 22.4%

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

10. Simplified 22.4%

    \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}} \]
11. Step-by-step derivation

    1. sqrt-unprod 37.8%

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

    2. *-commutative 37.8%

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

    3. *-commutative 37.8%

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

    4. associate-*r* 37.8%

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

    5. associate-*r* 36.3%

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

12. Applied egg-rr 36.3%

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

13. Final simplification 36.3%

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

Alternative 15: 34.9% 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 52.3%

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

3. Simplified 59.4%

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

4. Taylor expanded in t around inf 37.8%

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

5. Final simplification 37.8%

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

Reproduce

herbie shell --seed 2023215 
(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*))))))