Toniolo and Linder, Equation (13)

Percentage Accurate: 49.2% → 67.9%

Time: 22.3s

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: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 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 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))))
     (if (<= t_1 5e+151)
       t_1
       (*
        (sqrt 2.0)
        (*
         l
         (sqrt (* n (* U (+ (* (/ n Om) (/ (- 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 = sqrt((((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 * (n * (U * (t + (-2.0 * (l * (l / Om))))))));
	} else if (t_1 <= 5e+151) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n * (U * (((n / Om) * ((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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((2.0d0 * (n * (u * (t + ((-2.0d0) * (l * (l / om))))))))
    else if (t_1 <= 5d+151) then
        tmp = t_1
    else
        tmp = sqrt(2.0d0) * (l * sqrt((n * (u * (((n / om) * ((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 t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))));
	} else if (t_1 <= 5e+151) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((n * (U * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))))
	elif t_1 <= 5e+151:
		tmp = t_1
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((n * (U * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(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(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))));
	elseif (t_1 <= 5e+151)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))));
	elseif (t_1 <= 5e+151)
		tmp = t_1;
	else
		tmp = sqrt(2.0) * (l * sqrt((n * (U * (((n / Om) * ((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$42$_] := Block[{t$95$1 = 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+151], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n * N[(U * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.2%

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

   3. Simplified 9.9%

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

      1. associate-*l/ 9.9%

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

   5. Applied egg-rr 9.9%

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

   6. Taylor expanded in n around 0 50.8%

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

      1. *-commutative 50.8%

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

      2. unpow2 50.8%

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

      3. associate-*r/ 50.8%

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

   8. Simplified 50.8%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.0000000000000002e151

   1. Initial program 97.0%

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

   if 5.0000000000000002e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 21.4%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in l around inf 23.8%

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

      1. associate-*l* 23.8%

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

      2. associate-*r* 23.8%

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

      3. sub-neg 23.8%

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

      4. associate-/l* 24.0%

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

      5. unpow2 24.0%

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

      6. associate-*r/ 24.0%

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

      7. metadata-eval 24.0%

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

      8. distribute-neg-frac 24.0%

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

      9. metadata-eval 24.0%

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

   6. Simplified 24.0%

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

      1. *-un-lft-identity 24.0%

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

      2. associate-*l* 24.0%

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

      3. associate-/l* 25.9%

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

   8. Applied egg-rr 25.9%

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

      1. *-lft-identity 25.9%

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

      2. associate-/r/ 26.7%

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

   10. Simplified 26.7%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 2: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.9e+31)
   (pow (* 2.0 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))) 0.5)
   (*
    (sqrt 2.0)
    (* l (sqrt (* n (* U (+ (* (/ n Om) (/ (- 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 <= 3.9e+31) {
		tmp = pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n * (U * (((n / Om) * ((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 <= 3.9d+31) then
        tmp = (2.0d0 * (n * (u * (t + ((-2.0d0) * (l * (l / om))))))) ** 0.5d0
    else
        tmp = sqrt(2.0d0) * (l * sqrt((n * (u * (((n / om) * ((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 <= 3.9e+31) {
		tmp = Math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5);
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((n * (U * (((n / Om) * ((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 <= 3.9e+31:
		tmp = math.pow((2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))), 0.5)
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt((n * (U * (((n / Om) * ((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 <= 3.9e+31)
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))))) ^ 0.5;
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + 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 <= 3.9e+31)
		tmp = (2.0 * (n * (U * (t + (-2.0 * (l * (l / Om))))))) ^ 0.5;
	else
		tmp = sqrt(2.0) * (l * sqrt((n * (U * (((n / Om) * ((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, 3.9e+31], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n * N[(U * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.89999999999999999e31

   1. Initial program 49.9%

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

   3. Simplified 48.4%

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

      1. associate-*l/ 50.4%

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

   5. Applied egg-rr 50.4%

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

   6. Taylor expanded in n around 0 48.0%

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

      1. *-commutative 48.0%

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

      2. unpow2 48.0%

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

      3. associate-*r/ 49.0%

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

   8. Simplified 49.0%

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

      1. pow1/2 52.0%

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

   10. Applied egg-rr 52.0%

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

   if 3.89999999999999999e31 < l

   1. Initial program 38.7%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in l around inf 61.9%

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

      1. associate-*l* 62.0%

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

      2. associate-*r* 62.0%

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

      3. sub-neg 62.0%

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

      4. associate-/l* 62.5%

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

      5. unpow2 62.5%

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

      6. associate-*r/ 62.5%

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

      7. metadata-eval 62.5%

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

      8. distribute-neg-frac 62.5%

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

      9. metadata-eval 62.5%

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

   6. Simplified 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. associate-*l* 62.5%

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

      3. associate-/l* 67.2%

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

   8. Applied egg-rr 67.2%

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

      1. *-lft-identity 67.2%

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

      2. associate-/r/ 70.7%

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

   10. Simplified 70.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)}\right)\\ \end{array} \]

Alternative 3: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\\ \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 2.36 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t_1 + \frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}}\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (* l (/ l Om))))))
   (if (<= l 4.2e-163)
     (sqrt (* (* 2.0 n) (* U t)))
     (if (<= l 2.36e-23)
       (sqrt (* (* 2.0 (* n U)) (+ t_1 (/ n (* (/ Om (* l l)) (/ Om U*))))))
       (if (<= l 3.4e+31)
         (pow (* 2.0 (* n (* U t_1))) 0.5)
         (if (<= l 2.2e+143)
           (sqrt
            (*
             -2.0
             (* (* n (* l l)) (* U (- (/ 2.0 Om) (* (/ n Om) (/ U* Om)))))))
           (* l (* (sqrt 2.0) (sqrt (* -2.0 (/ U (/ Om n))))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l * (l / Om)));
	double tmp;
	if (l <= 4.2e-163) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 2.36e-23) {
		tmp = sqrt(((2.0 * (n * U)) * (t_1 + (n / ((Om / (l * l)) * (Om / U_42_))))));
	} else if (l <= 3.4e+31) {
		tmp = pow((2.0 * (n * (U * t_1))), 0.5);
	} else if (l <= 2.2e+143) {
		tmp = sqrt((-2.0 * ((n * (l * l)) * (U * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	} else {
		tmp = l * (sqrt(2.0) * sqrt((-2.0 * (U / (Om / n)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l * (l / om)))
    if (l <= 4.2d-163) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else if (l <= 2.36d-23) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t_1 + (n / ((om / (l * l)) * (om / u_42))))))
    else if (l <= 3.4d+31) then
        tmp = (2.0d0 * (n * (u * t_1))) ** 0.5d0
    else if (l <= 2.2d+143) then
        tmp = sqrt(((-2.0d0) * ((n * (l * l)) * (u * ((2.0d0 / om) - ((n / om) * (u_42 / om)))))))
    else
        tmp = l * (sqrt(2.0d0) * sqrt(((-2.0d0) * (u / (om / n)))))
    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 = t + (-2.0 * (l * (l / Om)));
	double tmp;
	if (l <= 4.2e-163) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 2.36e-23) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t_1 + (n / ((Om / (l * l)) * (Om / U_42_))))));
	} else if (l <= 3.4e+31) {
		tmp = Math.pow((2.0 * (n * (U * t_1))), 0.5);
	} else if (l <= 2.2e+143) {
		tmp = Math.sqrt((-2.0 * ((n * (l * l)) * (U * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	} else {
		tmp = l * (Math.sqrt(2.0) * Math.sqrt((-2.0 * (U / (Om / n)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l * (l / Om)))
	tmp = 0
	if l <= 4.2e-163:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	elif l <= 2.36e-23:
		tmp = math.sqrt(((2.0 * (n * U)) * (t_1 + (n / ((Om / (l * l)) * (Om / U_42_))))))
	elif l <= 3.4e+31:
		tmp = math.pow((2.0 * (n * (U * t_1))), 0.5)
	elif l <= 2.2e+143:
		tmp = math.sqrt((-2.0 * ((n * (l * l)) * (U * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))))
	else:
		tmp = l * (math.sqrt(2.0) * math.sqrt((-2.0 * (U / (Om / n)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om))))
	tmp = 0.0
	if (l <= 4.2e-163)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (l <= 2.36e-23)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t_1 + Float64(n / Float64(Float64(Om / Float64(l * l)) * Float64(Om / U_42_))))));
	elseif (l <= 3.4e+31)
		tmp = Float64(2.0 * Float64(n * Float64(U * t_1))) ^ 0.5;
	elseif (l <= 2.2e+143)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * Float64(l * l)) * Float64(U * Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(U_42_ / Om)))))));
	else
		tmp = Float64(l * Float64(sqrt(2.0) * sqrt(Float64(-2.0 * Float64(U / Float64(Om / n))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l * (l / Om)));
	tmp = 0.0;
	if (l <= 4.2e-163)
		tmp = sqrt(((2.0 * n) * (U * t)));
	elseif (l <= 2.36e-23)
		tmp = sqrt(((2.0 * (n * U)) * (t_1 + (n / ((Om / (l * l)) * (Om / U_42_))))));
	elseif (l <= 3.4e+31)
		tmp = (2.0 * (n * (U * t_1))) ^ 0.5;
	elseif (l <= 2.2e+143)
		tmp = sqrt((-2.0 * ((n * (l * l)) * (U * ((2.0 / Om) - ((n / Om) * (U_42_ / Om)))))));
	else
		tmp = l * (sqrt(2.0) * sqrt((-2.0 * (U / (Om / n)))));
	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[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.2e-163], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.36e-23], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[(n / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.4e+31], N[Power[N[(2.0 * N[(n * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.2e+143], N[Sqrt[N[(-2.0 * N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 4.19999999999999996e-163

   1. Initial program 49.2%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in l around 0 37.5%

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

   5. Taylor expanded in n around 0 42.0%

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

      1. associate-*r* 42.0%

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

      2. *-commutative 42.0%

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

   7. Simplified 42.0%

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

   if 4.19999999999999996e-163 < l < 2.3600000000000001e-23

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. associate-*l/ 64.2%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in U around 0 60.8%

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

      1. mul-1-neg 60.8%

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

      2. associate-/l* 60.8%

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

      3. distribute-neg-frac 60.8%

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

      4. unpow2 60.8%

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

      5. times-frac 67.7%

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

      6. unpow2 67.7%

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

   8. Simplified 67.7%

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

   if 2.3600000000000001e-23 < l < 3.3999999999999998e31

   1. Initial program 33.3%

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

   3. Simplified 33.0%

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

      1. associate-*l/ 32.9%

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

   5. Applied egg-rr 32.9%

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

   6. Taylor expanded in n around 0 32.7%

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

      1. *-commutative 32.7%

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

      2. unpow2 32.7%

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

      3. associate-*r/ 32.7%

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

   8. Simplified 32.7%

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

      1. pow1/2 38.9%

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

   10. Applied egg-rr 38.9%

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

   if 3.3999999999999998e31 < l < 2.20000000000000014e143

   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 49.6%

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

   4. Taylor expanded in l around inf 43.9%

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

      1. *-commutative 43.9%

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

      2. *-commutative 43.9%

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

      3. unpow2 43.9%

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

      4. +-commutative 43.9%

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

      5. associate-*r/ 43.9%

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

      6. metadata-eval 43.9%

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

      7. associate-/l* 43.9%

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

      8. unpow2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in U around 0 44.2%

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

      1. associate-*r/ 44.2%

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

      2. mul-1-neg 44.2%

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

      3. unpow2 44.2%

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

   9. Simplified 44.2%

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

   10. Taylor expanded in U around 0 47.4%

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

       1. associate-*r* 58.5%

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

       2. unpow2 58.5%

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

       3. *-commutative 58.5%

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

       4. associate-*r/ 58.5%

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

       5. metadata-eval 58.5%

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

       6. +-commutative 58.5%

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

       7. mul-1-neg 58.5%

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

       8. unsub-neg 58.5%

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

       9. unpow2 58.5%

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

       10. times-frac 58.5%

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

   12. Simplified 58.5%

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

   if 2.20000000000000014e143 < l

   1. Initial program 33.6%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in l around inf 65.2%

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

      1. associate-*l* 65.3%

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

      2. associate-*r* 65.2%

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

      3. sub-neg 65.2%

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

      4. associate-/l* 66.0%

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

      5. unpow2 66.0%

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

      6. associate-*r/ 66.0%

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

      7. metadata-eval 66.0%

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

      8. distribute-neg-frac 66.0%

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

      9. metadata-eval 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in n around 0 47.4%

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

      1. associate-*r/ 47.4%

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

      2. associate-*l/ 47.4%

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

      3. *-commutative 47.4%

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

   9. Simplified 47.4%

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

       1. pow1 47.4%

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

       2. *-commutative 47.4%

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

       3. associate-*l* 50.0%

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

   11. Applied egg-rr 50.0%

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

       1. unpow1 50.0%

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

       2. associate-*l* 50.0%

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

       3. associate-*r* 47.4%

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

       4. associate-*r/ 47.4%

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

       5. associate-*l/ 47.4%

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

       6. *-commutative 47.4%

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

       7. *-commutative 47.4%

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

       8. associate-/l* 55.4%

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

   13. Simplified 55.4%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 2.36 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \frac{n}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{U*}}\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U*}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{2} \cdot \sqrt{-2 \cdot \frac{U}{\frac{Om}{n}}}\right)\\ \end{array} \]

Alternative 4: 41.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.2e+28)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= l 8.8e+150)
     (sqrt (* -2.0 (* 2.0 (/ (* (* l l) (* n U)) Om))))
     (sqrt (* -2.0 (* n (* U (* 2.0 (* l (/ l Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+28) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 8.8e+150) {
		tmp = sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	} else {
		tmp = sqrt((-2.0 * (n * (U * (2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.2d+28) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 8.8d+150) then
        tmp = sqrt(((-2.0d0) * (2.0d0 * (((l * l) * (n * u)) / om))))
    else
        tmp = sqrt(((-2.0d0) * (n * (u * (2.0d0 * (l * (l / om)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.2e+28) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 8.8e+150) {
		tmp = Math.sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	} else {
		tmp = Math.sqrt((-2.0 * (n * (U * (2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.2e+28:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 8.8e+150:
		tmp = math.sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))))
	else:
		tmp = math.sqrt((-2.0 * (n * (U * (2.0 * (l * (l / Om)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.2e+28)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 8.8e+150)
		tmp = sqrt(Float64(-2.0 * Float64(2.0 * Float64(Float64(Float64(l * l) * Float64(n * U)) / Om))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(n * Float64(U * Float64(2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.2e+28)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 8.8e+150)
		tmp = sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	else
		tmp = sqrt((-2.0 * (n * (U * (2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.2e+28], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 8.8e+150], N[Sqrt[N[(-2.0 * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(n * N[(U * N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.2000000000000001e28

   1. Initial program 49.9%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in l around 0 41.9%

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

      1. pow1/2 42.9%

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

      2. *-commutative 42.9%

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

   6. Applied egg-rr 42.9%

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

   if 6.2000000000000001e28 < l < 8.79999999999999998e150

   1. Initial program 48.4%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in l around inf 40.8%

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

      1. *-commutative 40.8%

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

      2. *-commutative 40.8%

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

      3. unpow2 40.8%

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

      4. +-commutative 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. associate-/l* 40.9%

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

      8. unpow2 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in n around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. associate-*r* 43.9%

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

      3. unpow2 43.9%

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

   9. Simplified 43.9%

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

   if 8.79999999999999998e150 < l

   1. Initial program 33.6%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in l around inf 37.7%

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

      1. *-commutative 37.7%

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

      2. *-commutative 37.7%

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

      3. unpow2 37.7%

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

      4. +-commutative 37.7%

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

      5. associate-*r/ 37.7%

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

      6. metadata-eval 37.7%

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

      7. associate-/l* 43.7%

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

      8. unpow2 43.7%

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

   6. Simplified 43.7%

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

   7. Taylor expanded in U around 0 37.2%

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

      1. associate-*r* 34.8%

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

      2. unpow2 34.8%

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

      3. +-commutative 34.8%

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

      4. mul-1-neg 34.8%

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

      5. unsub-neg 34.8%

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

      6. associate-*r/ 34.8%

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

      7. metadata-eval 34.8%

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

      8. unpow2 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in Om around inf 34.9%

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

       1. unpow2 34.9%

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

       2. associate-*r/ 43.4%

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

   12. Simplified 43.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(U \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 40.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.2e+28)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= l 1.04e+152)
     (sqrt (* -2.0 (* 2.0 (/ (* (* l l) (* n U)) Om))))
     (sqrt (* -2.0 (* n (* (/ 2.0 Om) (* l (* U l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.2e+28) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 1.04e+152) {
		tmp = sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	} else {
		tmp = sqrt((-2.0 * (n * ((2.0 / Om) * (l * (U * l))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.2d+28) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 1.04d+152) then
        tmp = sqrt(((-2.0d0) * (2.0d0 * (((l * l) * (n * u)) / om))))
    else
        tmp = sqrt(((-2.0d0) * (n * ((2.0d0 / om) * (l * (u * l))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.2e+28) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 1.04e+152) {
		tmp = Math.sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	} else {
		tmp = Math.sqrt((-2.0 * (n * ((2.0 / Om) * (l * (U * l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.2e+28:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 1.04e+152:
		tmp = math.sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))))
	else:
		tmp = math.sqrt((-2.0 * (n * ((2.0 / Om) * (l * (U * l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.2e+28)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 1.04e+152)
		tmp = sqrt(Float64(-2.0 * Float64(2.0 * Float64(Float64(Float64(l * l) * Float64(n * U)) / Om))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(n * Float64(Float64(2.0 / Om) * Float64(l * Float64(U * l))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.2e+28)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 1.04e+152)
		tmp = sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	else
		tmp = sqrt((-2.0 * (n * ((2.0 / Om) * (l * (U * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.2e+28], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.04e+152], N[Sqrt[N[(-2.0 * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(n * N[(N[(2.0 / Om), $MachinePrecision] * N[(l * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.19999999999999978e28

   1. Initial program 49.9%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in l around 0 41.9%

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

      1. pow1/2 42.9%

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

      2. *-commutative 42.9%

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

   6. Applied egg-rr 42.9%

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

   if 4.19999999999999978e28 < l < 1.04000000000000005e152

   1. Initial program 48.4%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in l around inf 40.8%

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

      1. *-commutative 40.8%

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

      2. *-commutative 40.8%

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

      3. unpow2 40.8%

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

      4. +-commutative 40.8%

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

      5. associate-*r/ 40.8%

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

      6. metadata-eval 40.8%

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

      7. associate-/l* 40.9%

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

      8. unpow2 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in n around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. associate-*r* 43.9%

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

      3. unpow2 43.9%

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

   9. Simplified 43.9%

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

   if 1.04000000000000005e152 < l

   1. Initial program 33.6%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in l around inf 37.7%

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

      1. *-commutative 37.7%

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

      2. *-commutative 37.7%

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

      3. unpow2 37.7%

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

      4. +-commutative 37.7%

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

      5. associate-*r/ 37.7%

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

      6. metadata-eval 37.7%

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

      7. associate-/l* 43.7%

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

      8. unpow2 43.7%

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

   6. Simplified 43.7%

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

   7. Taylor expanded in U around 0 37.2%

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

      1. associate-*r* 34.8%

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

      2. unpow2 34.8%

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

      3. +-commutative 34.8%

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

      4. mul-1-neg 34.8%

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

      5. unsub-neg 34.8%

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

      6. associate-*r/ 34.8%

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

      7. metadata-eval 34.8%

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

      8. unpow2 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in Om around inf 37.8%

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

       1. associate-*r/ 37.8%

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

       2. *-commutative 37.8%

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

       3. unpow2 37.8%

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

       4. associate-*l/ 37.8%

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

       5. associate-*r* 46.3%

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

   12. Simplified 46.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(n \cdot \left(\frac{2}{Om} \cdot \left(\ell \cdot \left(U \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 43.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.8)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (pow (* 2.0 (* n (* -2.0 (* U (/ (* l 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 <= 4.8) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = pow((2.0 * (n * (-2.0 * (U * ((l * 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 <= 4.8d0) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (n * ((-2.0d0) * (u * ((l * 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 <= 4.8) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (n * (-2.0 * (U * ((l * l) / Om))))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.8:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.pow((2.0 * (n * (-2.0 * (U * ((l * l) / Om))))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.8)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(n * Float64(-2.0 * Float64(U * Float64(Float64(l * 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 <= 4.8)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = (2.0 * (n * (-2.0 * (U * ((l * 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, 4.8], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(-2.0 * N[(U * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.79999999999999982

   1. Initial program 51.0%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in l around 0 43.1%

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

      1. pow1/2 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

   if 4.79999999999999982 < l

   1. Initial program 37.4%

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

   3. Simplified 35.8%

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

      1. associate-*l/ 41.9%

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

   5. Applied egg-rr 41.9%

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

   6. Taylor expanded in n around 0 33.9%

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

      1. *-commutative 33.9%

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

      2. unpow2 33.9%

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

      3. associate-*r/ 41.4%

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

   8. Simplified 41.4%

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

      1. pow1/2 48.0%

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

   10. Applied egg-rr 48.0%

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

   11. Taylor expanded in t around 0 38.6%

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

       1. associate-*l/ 38.7%

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

       2. unpow2 38.7%

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

       3. associate-*r/ 43.2%

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

       4. associate-*r/ 38.7%

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

   13. Simplified 38.7%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7: 42.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \frac{-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.3e+28)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (pow (* 2.0 (/ (* -2.0 (* (* l l) (* n U))) 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 <= 1.3e+28) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = pow((2.0 * ((-2.0 * ((l * l) * (n * U))) / 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 <= 1.3d+28) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (((-2.0d0) * ((l * l) * (n * u))) / 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 <= 1.3e+28) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * ((-2.0 * ((l * l) * (n * U))) / Om)), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.3e+28:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.pow((2.0 * ((-2.0 * ((l * l) * (n * U))) / Om)), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.3e+28)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(Float64(-2.0 * Float64(Float64(l * l) * Float64(n * U))) / 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 <= 1.3e+28)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = (2.0 * ((-2.0 * ((l * l) * (n * U))) / 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, 1.3e+28], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3000000000000001e28

   1. Initial program 49.9%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in l around 0 41.9%

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

      1. pow1/2 42.9%

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

      2. *-commutative 42.9%

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

   6. Applied egg-rr 42.9%

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

   if 1.3000000000000001e28 < l

   1. Initial program 39.2%

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

   3. Simplified 37.5%

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

      1. associate-*l/ 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in n around 0 35.1%

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

      1. *-commutative 35.1%

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

      2. unpow2 35.1%

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

      3. associate-*r/ 43.9%

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

   8. Simplified 43.9%

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

      1. pow1/2 51.5%

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

   10. Applied egg-rr 51.5%

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

   11. Taylor expanded in t around 0 42.6%

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

       1. associate-*r/ 42.6%

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

       2. *-commutative 42.6%

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

       3. associate-*r* 45.8%

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

       4. unpow2 45.8%

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

   13. Simplified 45.8%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \frac{-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)\right)}{Om}\right)}^{0.5}\\ \end{array} \]

Alternative 8: 52.2% accurate, 1.9× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (pow (* 2.0 (* n (* U (+ t (* -2.0 (* l (/ l Om))))))) 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 + (-2.0 * (l * (l / Om))))))), 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 + ((-2.0d0) * (l * (l / om))))))) ** 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 + (-2.0 * (l * (l / Om))))))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

3. Simplified 46.0%

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

   1. associate-*l/ 49.1%

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

5. Applied egg-rr 49.1%

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

6. Taylor expanded in n around 0 45.2%

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

   1. *-commutative 45.2%

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

   2. unpow2 45.2%

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

   3. associate-*r/ 47.9%

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

8. Simplified 47.9%

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

   1. pow1/2 51.9%

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

10. Applied egg-rr 51.9%

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

11. Final simplification 51.9%

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

Alternative 9: 38.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.8)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* -2.0 (* 2.0 (* (/ n Om) (* U (* l l))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.8) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (2.0 * ((n / Om) * (U * (l * l))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.8d0) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((-2.0d0) * (2.0d0 * ((n / om) * (u * (l * l))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.8) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (2.0 * ((n / Om) * (U * (l * l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.8:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (2.0 * ((n / Om) * (U * (l * l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.8)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(2.0 * Float64(Float64(n / Om) * Float64(U * Float64(l * l))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.8)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (2.0 * ((n / Om) * (U * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.8], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(2.0 * N[(N[(n / Om), $MachinePrecision] * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.79999999999999982

   1. Initial program 51.0%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in l around 0 43.1%

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

      1. pow1/2 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

   if 4.79999999999999982 < l

   1. Initial program 37.4%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in l around inf 36.8%

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

      1. *-commutative 36.8%

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

      2. *-commutative 36.8%

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

      3. unpow2 36.8%

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

      4. +-commutative 36.8%

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

      5. associate-*r/ 36.8%

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

      6. metadata-eval 36.8%

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

      7. associate-/l* 40.0%

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

      8. unpow2 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in U around 0 40.2%

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

      1. associate-*r/ 40.2%

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

      2. mul-1-neg 40.2%

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

      3. unpow2 40.2%

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

   9. Simplified 40.2%

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

   10. Taylor expanded in n around 0 31.9%

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

       1. associate-/l* 32.0%

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

       2. associate-/r/ 30.4%

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

       3. *-commutative 30.4%

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

       4. unpow2 30.4%

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

   12. Simplified 30.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \left(\frac{n}{Om} \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 10: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.8)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* -2.0 (* 2.0 (/ n (/ Om (* U (* l l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.8) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (2.0 * (n / (Om / (U * (l * l)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.8d0) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((-2.0d0) * (2.0d0 * (n / (om / (u * (l * l)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.8) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (2.0 * (n / (Om / (U * (l * l)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.8:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (2.0 * (n / (Om / (U * (l * l)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.8)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(2.0 * Float64(n / Float64(Om / Float64(U * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.8)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (2.0 * (n / (Om / (U * (l * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.8], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(2.0 * N[(n / N[(Om / N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.79999999999999982

   1. Initial program 51.0%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in l around 0 43.1%

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

      1. pow1/2 44.1%

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

      2. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

   if 4.79999999999999982 < l

   1. Initial program 37.4%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in l around inf 36.8%

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

      1. *-commutative 36.8%

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

      2. *-commutative 36.8%

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

      3. unpow2 36.8%

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

      4. +-commutative 36.8%

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

      5. associate-*r/ 36.8%

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

      6. metadata-eval 36.8%

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

      7. associate-/l* 40.0%

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

      8. unpow2 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in n around 0 31.9%

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

      1. associate-/l* 32.0%

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

      2. *-commutative 32.0%

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

      3. unpow2 32.0%

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

   9. Simplified 32.0%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{n}{\frac{Om}{U \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \end{array} \]

Alternative 11: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.2e+28)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* -2.0 (* 2.0 (/ (* (* l l) (* n U)) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.2e+28) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / 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 <= 4.2d+28) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((-2.0d0) * (2.0d0 * (((l * l) * (n * u)) / 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 <= 4.2e+28) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.2e+28:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / Om))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.2e+28)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(2.0 * Float64(Float64(Float64(l * l) * Float64(n * U)) / 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 <= 4.2e+28)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (2.0 * (((l * l) * (n * U)) / 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, 4.2e+28], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.19999999999999978e28

   1. Initial program 49.9%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in l around 0 41.9%

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

      1. pow1/2 42.9%

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

      2. *-commutative 42.9%

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

   6. Applied egg-rr 42.9%

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

   if 4.19999999999999978e28 < l

   1. Initial program 39.2%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in l around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. *-commutative 38.9%

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

      3. unpow2 38.9%

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

      4. +-commutative 38.9%

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

      5. associate-*r/ 38.9%

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

      6. metadata-eval 38.9%

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

      7. associate-/l* 42.6%

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

      8. unpow2 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in n around 0 34.9%

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

      1. *-commutative 34.9%

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

      2. associate-*r* 38.2%

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

      3. unpow2 38.2%

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

   9. Simplified 38.2%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+28}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot U\right)}{Om}\right)}\\ \end{array} \]

Alternative 12: 46.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

3. Simplified 46.0%

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

   1. associate-*l/ 49.1%

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

5. Applied egg-rr 49.1%

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

6. Taylor expanded in n around 0 45.2%

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

   1. *-commutative 45.2%

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

   2. unpow2 45.2%

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

   3. associate-*r/ 47.9%

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

8. Simplified 47.9%

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

9. Final simplification 47.9%

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

Alternative 13: 36.8% accurate, 2.1× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* 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(2.0 * Float64(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[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 47.6%

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

3. Simplified 49.1%

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

4. Taylor expanded in l around 0 34.9%

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

   1. pow1/2 36.0%

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

   2. *-commutative 36.0%

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

6. Applied egg-rr 36.0%

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

7. Final simplification 36.0%

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

Alternative 14: 35.3% accurate, 2.1× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* 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(2.0 * Float64(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[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 47.6%

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

3. Simplified 49.1%

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

4. Taylor expanded in l around 0 34.9%

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

5. Final simplification 34.9%

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

Alternative 15: 35.3% 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 47.6%

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

3. Simplified 49.1%

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

4. Taylor expanded in l around 0 31.4%

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

5. Taylor expanded in n around 0 34.9%

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

   1. associate-*r* 34.9%

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

   2. *-commutative 34.9%

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

7. Simplified 34.9%

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

8. Final simplification 34.9%

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

Reproduce

herbie shell --seed 2023263 
(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*))))))