Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 68.5%

Time: 42.2s

Alternatives: 21

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: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := U \cdot \left(2 \cdot n\right)\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot t_1\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* U (* 2.0 n)))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 0.0)
     (sqrt (* 2.0 (* U (* n (fma t_1 -2.0 t)))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (*
        (* l (sqrt 2.0))
        (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = U * (2.0 * n);
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * fma(t_1, -2.0, t)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(U * Float64(2.0 * n))
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(t_1, -2.0, t)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U)))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $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 9.4%

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

      1. associate-*l/ 9.4%

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

   3. Applied egg-rr 9.4%

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

   4. Taylor expanded in n around 0 40.0%

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

      1. cancel-sign-sub-inv 40.0%

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

      2. metadata-eval 40.0%

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

      3. +-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. unpow2 40.0%

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

      6. associate-*r/ 40.0%

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

      7. fma-udef 40.0%

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

   6. Simplified 40.0%

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

   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*))))) < +inf.0

   1. Initial program 65.0%

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

      1. associate-*l/ 70.9%

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

   3. Applied egg-rr 70.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in l around inf 28.8%

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

      1. unpow2 28.8%

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

      2. sub-neg 28.8%

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

      3. unpow2 28.8%

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

      4. times-frac 37.0%

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

      5. associate-*r/ 37.0%

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

      6. metadata-eval 37.0%

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

      7. distribute-neg-frac 37.0%

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

      8. metadata-eval 37.0%

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

   6. Simplified 37.0%

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

   7. Taylor expanded in l around 0 29.3%

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

      1. *-commutative 29.3%

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

      2. sub-neg 29.3%

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

      3. unpow2 29.3%

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

      4. times-frac 36.8%

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

      5. distribute-rgt-in 31.2%

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

      6. associate-*r/ 31.2%

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

      7. metadata-eval 31.2%

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

      8. distribute-neg-frac 31.2%

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

      9. metadata-eval 31.2%

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

   9. Simplified 40.3%

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

4. Final simplification 61.5%

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

Alternative 2: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + t_2 \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-18}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(t_2, -2, t\right) + \frac{U*}{\frac{Om \cdot Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+50}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.32 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+192}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0)))
        (t_2 (/ (* l l) Om))
        (t_3
         (sqrt
          (*
           (* 2.0 (* n U))
           (+ (+ t (* t_2 -2.0)) (* n (* (pow (/ l Om) 2.0) (- U* U))))))))
   (if (<= l 1.18e-18)
     t_3
     (if (<= l 8.5e+24)
       (sqrt
        (*
         2.0
         (* U (* n (+ (fma t_2 -2.0 t) (/ U* (/ (* Om Om) (* n (* l l)))))))))
       (if (<= l 4.4e+50)
         t_3
         (if (<= l 1.32e+132)
           (sqrt
            (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
           (if (<= l 3.8e+192)
             (* t_1 (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U)))))))
             (if (<= l 2.5e+243)
               (sqrt
                (*
                 2.0
                 (*
                  (* U l)
                  (* (* n l) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))))
               (*
                t_1
                (sqrt
                 (/ (* U (* n (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double t_2 = (l * l) / Om;
	double t_3 = sqrt(((2.0 * (n * U)) * ((t + (t_2 * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	double tmp;
	if (l <= 1.18e-18) {
		tmp = t_3;
	} else if (l <= 8.5e+24) {
		tmp = sqrt((2.0 * (U * (n * (fma(t_2, -2.0, t) + (U_42_ / ((Om * Om) / (n * (l * l)))))))));
	} else if (l <= 4.4e+50) {
		tmp = t_3;
	} else if (l <= 1.32e+132) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 3.8e+192) {
		tmp = t_1 * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	} else if (l <= 2.5e+243) {
		tmp = sqrt((2.0 * ((U * l) * ((n * l) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om))))));
	} else {
		tmp = t_1 * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t + Float64(t_2 * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))))
	tmp = 0.0
	if (l <= 1.18e-18)
		tmp = t_3;
	elseif (l <= 8.5e+24)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(fma(t_2, -2.0, t) + Float64(U_42_ / Float64(Float64(Om * Om) / Float64(n * Float64(l * l)))))))));
	elseif (l <= 4.4e+50)
		tmp = t_3;
	elseif (l <= 1.32e+132)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l <= 3.8e+192)
		tmp = Float64(t_1 * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U)))))));
	elseif (l <= 2.5e+243)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * l) * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om))))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t + N[(t$95$2 * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1.18e-18], t$95$3, If[LessEqual[l, 8.5e+24], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(t$95$2 * -2.0 + t), $MachinePrecision] + N[(U$42$ / N[(N[(Om * Om), $MachinePrecision] / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.4e+50], t$95$3, If[LessEqual[l, 1.32e+132], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e+192], N[(t$95$1 * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e+243], N[Sqrt[N[(2.0 * N[(N[(U * l), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < 1.18000000000000008e-18 or 8.49999999999999959e24 < l < 4.40000000000000034e50

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

   if 1.18000000000000008e-18 < l < 8.49999999999999959e24

   1. Initial program 53.5%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in U around 0 67.5%

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

      1. remove-double-neg 67.5%

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

      2. mul-1-neg 67.5%

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

      3. sub-neg 67.5%

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

      4. associate--l+ 67.5%

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

      5. sub-neg 67.5%

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

   6. Simplified 78.6%

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

   if 4.40000000000000034e50 < l < 1.3199999999999999e132

   1. Initial program 50.7%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in Om around inf 61.8%

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

   if 1.3199999999999999e132 < l < 3.7999999999999999e192

   1. Initial program 29.8%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in l around inf 31.9%

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

      1. unpow2 31.9%

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

      2. sub-neg 31.9%

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

      3. unpow2 31.9%

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

      4. times-frac 32.3%

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

      5. associate-*r/ 32.3%

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

      6. metadata-eval 32.3%

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

      7. distribute-neg-frac 32.3%

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

      8. metadata-eval 32.3%

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

   6. Simplified 32.3%

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

   7. Taylor expanded in l around 0 42.0%

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

      1. *-commutative 42.0%

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

      2. sub-neg 42.0%

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

      3. unpow2 42.0%

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

      4. times-frac 51.7%

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

      5. distribute-rgt-in 51.7%

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

      6. associate-*r/ 51.7%

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

      7. metadata-eval 51.7%

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

      8. distribute-neg-frac 51.7%

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

      9. metadata-eval 51.7%

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

   9. Simplified 70.1%

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

   if 3.7999999999999999e192 < l < 2.50000000000000019e243

   1. Initial program 22.1%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in l around inf 36.8%

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

      1. unpow2 36.8%

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

      2. sub-neg 36.8%

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

      3. unpow2 36.8%

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

      4. times-frac 37.3%

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

      5. associate-*r/ 37.3%

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

      6. metadata-eval 37.3%

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

      7. distribute-neg-frac 37.3%

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

      8. metadata-eval 37.3%

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

   6. Simplified 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. associate-*l* 58.9%

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

      3. fma-def 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. *-lft-identity 58.9%

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

      2. associate-*r* 65.8%

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

      3. associate-*r* 92.4%

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

   10. Simplified 92.4%

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

   if 2.50000000000000019e243 < l

   1. Initial program 14.0%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in l around inf 28.4%

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

      1. unpow2 28.4%

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

      2. sub-neg 28.4%

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

      3. unpow2 28.4%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 34.7%

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

      10. fma-udef 34.7%

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

      11. distribute-rgt-out 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in l around 0 72.5%

       \[\leadsto \color{blue}{\sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) + \frac{U*}{\frac{Om \cdot Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.32 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+192}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 3: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) + \frac{U*}{\frac{Om \cdot Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+193}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.46 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0))))
   (if (<= l 1.5e+29)
     (sqrt
      (*
       2.0
       (*
        U
        (*
         n
         (+ (fma (/ (* l l) Om) -2.0 t) (/ U* (/ (* Om Om) (* n (* l l)))))))))
     (if (<= l 9e+130)
       (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
       (if (<= l 3.3e+193)
         (* t_1 (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U)))))))
         (if (<= l 1.46e+245)
           (sqrt
            (*
             2.0
             (*
              (* U l)
              (* (* n l) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))))
           (*
            t_1
            (sqrt (/ (* U (* n (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double tmp;
	if (l <= 1.5e+29) {
		tmp = sqrt((2.0 * (U * (n * (fma(((l * l) / Om), -2.0, t) + (U_42_ / ((Om * Om) / (n * (l * l)))))))));
	} else if (l <= 9e+130) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 3.3e+193) {
		tmp = t_1 * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	} else if (l <= 1.46e+245) {
		tmp = sqrt((2.0 * ((U * l) * ((n * l) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om))))));
	} else {
		tmp = t_1 * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (l <= 1.5e+29)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(fma(Float64(Float64(l * l) / Om), -2.0, t) + Float64(U_42_ / Float64(Float64(Om * Om) / Float64(n * Float64(l * l)))))))));
	elseif (l <= 9e+130)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l <= 3.3e+193)
		tmp = Float64(t_1 * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U)))))));
	elseif (l <= 1.46e+245)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * l) * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om))))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.5e+29], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0 + t), $MachinePrecision] + N[(U$42$ / N[(N[(Om * Om), $MachinePrecision] / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9e+130], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.3e+193], N[(t$95$1 * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.46e+245], N[Sqrt[N[(2.0 * N[(N[(U * l), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 1.5e29

   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 55.2%

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

   4. Taylor expanded in U around 0 45.8%

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

      1. remove-double-neg 45.8%

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

      2. mul-1-neg 45.8%

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

      3. sub-neg 45.8%

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

      4. associate--l+ 45.8%

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

      5. sub-neg 45.8%

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

   6. Simplified 47.4%

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

   if 1.5e29 < l < 9.00000000000000078e130

   1. Initial program 49.3%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in Om around inf 53.8%

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

   if 9.00000000000000078e130 < l < 3.3e193

   1. Initial program 29.8%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in l around inf 31.9%

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

      1. unpow2 31.9%

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

      2. sub-neg 31.9%

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

      3. unpow2 31.9%

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

      4. times-frac 32.3%

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

      5. associate-*r/ 32.3%

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

      6. metadata-eval 32.3%

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

      7. distribute-neg-frac 32.3%

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

      8. metadata-eval 32.3%

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

   6. Simplified 32.3%

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

   7. Taylor expanded in l around 0 42.0%

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

      1. *-commutative 42.0%

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

      2. sub-neg 42.0%

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

      3. unpow2 42.0%

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

      4. times-frac 51.7%

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

      5. distribute-rgt-in 51.7%

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

      6. associate-*r/ 51.7%

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

      7. metadata-eval 51.7%

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

      8. distribute-neg-frac 51.7%

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

      9. metadata-eval 51.7%

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

   9. Simplified 70.1%

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

   if 3.3e193 < l < 1.45999999999999991e245

   1. Initial program 22.1%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in l around inf 36.8%

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

      1. unpow2 36.8%

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

      2. sub-neg 36.8%

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

      3. unpow2 36.8%

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

      4. times-frac 37.3%

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

      5. associate-*r/ 37.3%

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

      6. metadata-eval 37.3%

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

      7. distribute-neg-frac 37.3%

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

      8. metadata-eval 37.3%

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

   6. Simplified 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. associate-*l* 58.9%

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

      3. fma-def 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. *-lft-identity 58.9%

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

      2. associate-*r* 65.8%

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

      3. associate-*r* 92.4%

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

   10. Simplified 92.4%

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

   if 1.45999999999999991e245 < l

   1. Initial program 14.0%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in l around inf 28.4%

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

      1. unpow2 28.4%

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

      2. sub-neg 28.4%

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

      3. unpow2 28.4%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 34.7%

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

      10. fma-udef 34.7%

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

      11. distribute-rgt-out 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in l around 0 72.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) + \frac{U*}{\frac{Om \cdot Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+193}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.46 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 4: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq 9 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+193}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0))))
   (if (<= l 9e+129)
     (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
     (if (<= l 3e+193)
       (* t_1 (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U)))))))
       (if (<= l 8.8e+243)
         (sqrt
          (*
           2.0
           (* (* U l) (* (* n l) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))))
         (* t_1 (sqrt (/ (* U (* n (- (/ (* n (- U* U)) Om) 2.0))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double tmp;
	if (l <= 9e+129) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 3e+193) {
		tmp = t_1 * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	} else if (l <= 8.8e+243) {
		tmp = sqrt((2.0 * ((U * l) * ((n * l) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om))))));
	} else {
		tmp = t_1 * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (l <= 9e+129)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l <= 3e+193)
		tmp = Float64(t_1 * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U)))))));
	elseif (l <= 8.8e+243)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * l) * Float64(Float64(n * l) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om))))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 9e+129], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3e+193], N[(t$95$1 * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.8e+243], N[Sqrt[N[(2.0 * N[(N[(U * l), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 9.0000000000000003e129

   1. Initial program 50.8%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in Om around inf 45.5%

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

   if 9.0000000000000003e129 < l < 3e193

   1. Initial program 29.8%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in l around inf 31.9%

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

      1. unpow2 31.9%

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

      2. sub-neg 31.9%

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

      3. unpow2 31.9%

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

      4. times-frac 32.3%

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

      5. associate-*r/ 32.3%

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

      6. metadata-eval 32.3%

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

      7. distribute-neg-frac 32.3%

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

      8. metadata-eval 32.3%

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

   6. Simplified 32.3%

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

   7. Taylor expanded in l around 0 42.0%

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

      1. *-commutative 42.0%

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

      2. sub-neg 42.0%

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

      3. unpow2 42.0%

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

      4. times-frac 51.7%

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

      5. distribute-rgt-in 51.7%

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

      6. associate-*r/ 51.7%

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

      7. metadata-eval 51.7%

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

      8. distribute-neg-frac 51.7%

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

      9. metadata-eval 51.7%

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

   9. Simplified 70.1%

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

   if 3e193 < l < 8.80000000000000035e243

   1. Initial program 22.1%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in l around inf 36.8%

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

      1. unpow2 36.8%

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

      2. sub-neg 36.8%

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

      3. unpow2 36.8%

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

      4. times-frac 37.3%

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

      5. associate-*r/ 37.3%

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

      6. metadata-eval 37.3%

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

      7. distribute-neg-frac 37.3%

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

      8. metadata-eval 37.3%

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

   6. Simplified 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. associate-*l* 58.9%

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

      3. fma-def 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. *-lft-identity 58.9%

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

      2. associate-*r* 65.8%

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

      3. associate-*r* 92.4%

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

   10. Simplified 92.4%

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

   if 8.80000000000000035e243 < l

   1. Initial program 14.0%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in l around inf 28.4%

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

      1. unpow2 28.4%

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

      2. sub-neg 28.4%

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

      3. unpow2 28.4%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 34.7%

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

      10. fma-udef 34.7%

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

      11. distribute-rgt-out 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in l around 0 72.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+193}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+243}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 5: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.5e+133)
   (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
   (if (<= l 2.85e+206)
     (*
      (sqrt 2.0)
      (* l (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U))))))))
     (if (<= l 3.6e+244)
       (sqrt (* 2.0 (* U (* l (/ (* l (* n (- (/ (* n U*) Om) 2.0))) Om)))))
       (*
        (* l (sqrt 2.0))
        (sqrt (/ (* U (* n (- (/ (* n (- U* U)) Om) 2.0))) Om)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.5e+133) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 2.85e+206) {
		tmp = sqrt(2.0) * (l * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))));
	} else if (l <= 3.6e+244) {
		tmp = sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 3.5d+133) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else if (l <= 2.85d+206) then
        tmp = sqrt(2.0d0) * (l * sqrt((u * ((n / om) * ((-2.0d0) + ((n / om) * (u_42 - u)))))))
    else if (l <= 3.6d+244) then
        tmp = sqrt((2.0d0 * (u * (l * ((l * (n * (((n * u_42) / om) - 2.0d0))) / om)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((u * (n * (((n * (u_42 - u)) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.5e+133) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 2.85e+206) {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))));
	} else if (l <= 3.6e+244) {
		tmp = Math.sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.5e+133:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	elif l <= 2.85e+206:
		tmp = math.sqrt(2.0) * (l * math.sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))))
	elif l <= 3.6e+244:
		tmp = math.sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.5e+133)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l <= 2.85e+206)
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U))))))));
	elseif (l <= 3.6e+244)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(l * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.5e+133)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	elseif (l <= 2.85e+206)
		tmp = sqrt(2.0) * (l * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))));
	elseif (l <= 3.6e+244)
		tmp = sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.5e+133], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.85e+206], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6e+244], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(l * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.4999999999999998e133

   1. Initial program 50.8%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in Om around inf 45.5%

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

   if 3.4999999999999998e133 < l < 2.8499999999999999e206

   1. Initial program 29.0%

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

   3. Simplified 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. Step-by-step derivation

      1. sqrt-prod 50.4%

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

   5. Applied egg-rr 50.4%

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

   6. Taylor expanded in l around inf 44.5%

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

      1. *-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. unpow2 44.5%

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

      4. times-frac 58.5%

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

      5. distribute-rgt-in 58.5%

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

      6. associate-*r/ 58.5%

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

      7. metadata-eval 58.5%

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

      8. distribute-neg-frac 58.5%

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

      9. metadata-eval 58.5%

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

   8. Simplified 71.6%

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

   if 2.8499999999999999e206 < l < 3.6e244

   1. Initial program 20.3%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in l around inf 41.0%

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

      1. unpow2 41.0%

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

      2. sub-neg 41.0%

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

      3. unpow2 41.0%

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

      4. times-frac 41.0%

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

      5. associate-*r/ 41.0%

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

      6. metadata-eval 41.0%

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

      7. distribute-neg-frac 41.0%

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

      8. metadata-eval 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in l around 0 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. times-frac 41.0%

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

      4. fma-neg 41.0%

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

      5. associate-*r/ 41.0%

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

      6. metadata-eval 41.0%

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

      7. distribute-neg-frac 41.0%

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

      8. metadata-eval 41.0%

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

      9. associate-*r* 61.4%

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

      10. fma-udef 61.4%

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

      11. distribute-rgt-out 41.4%

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

   9. Simplified 61.4%

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

   10. Taylor expanded in U around 0 90.3%

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

   if 3.6e244 < l

   1. Initial program 14.0%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in l around inf 28.4%

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

      1. unpow2 28.4%

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

      2. sub-neg 28.4%

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

      3. unpow2 28.4%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 34.7%

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

      10. fma-udef 34.7%

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

      11. distribute-rgt-out 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in l around 0 72.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 6: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0))))
   (if (<= l 1.05e+130)
     (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
     (if (<= l 2.45e+207)
       (* t_1 (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U)))))))
       (if (<= l 2.5e+244)
         (sqrt (* 2.0 (* U (* l (/ (* l (* n (- (/ (* n U*) Om) 2.0))) Om)))))
         (* t_1 (sqrt (/ (* U (* n (- (/ (* n (- U* U)) Om) 2.0))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double tmp;
	if (l <= 1.05e+130) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 2.45e+207) {
		tmp = t_1 * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	} else if (l <= 2.5e+244) {
		tmp = sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	} else {
		tmp = t_1 * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * sqrt(2.0d0)
    if (l <= 1.05d+130) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else if (l <= 2.45d+207) then
        tmp = t_1 * sqrt((u * ((n / om) * ((-2.0d0) + ((n / om) * (u_42 - u))))))
    else if (l <= 2.5d+244) then
        tmp = sqrt((2.0d0 * (u * (l * ((l * (n * (((n * u_42) / om) - 2.0d0))) / om)))))
    else
        tmp = t_1 * sqrt(((u * (n * (((n * (u_42 - u)) / om) - 2.0d0))) / om))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * Math.sqrt(2.0);
	double tmp;
	if (l <= 1.05e+130) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 2.45e+207) {
		tmp = t_1 * Math.sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	} else if (l <= 2.5e+244) {
		tmp = Math.sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	} else {
		tmp = t_1 * Math.sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = l * math.sqrt(2.0)
	tmp = 0
	if l <= 1.05e+130:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	elif l <= 2.45e+207:
		tmp = t_1 * math.sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))))
	elif l <= 2.5e+244:
		tmp = math.sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))))
	else:
		tmp = t_1 * math.sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (l <= 1.05e+130)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l <= 2.45e+207)
		tmp = Float64(t_1 * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U)))))));
	elseif (l <= 2.5e+244)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(l * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * sqrt(2.0);
	tmp = 0.0;
	if (l <= 1.05e+130)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	elseif (l <= 2.45e+207)
		tmp = t_1 * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U))))));
	elseif (l <= 2.5e+244)
		tmp = sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	else
		tmp = t_1 * sqrt(((U * (n * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.05e+130], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.45e+207], N[(t$95$1 * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e+244], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(l * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.04999999999999995e130

   1. Initial program 50.8%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in Om around inf 45.5%

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

   if 1.04999999999999995e130 < l < 2.45e207

   1. Initial program 29.0%

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

   3. Simplified 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 inf 30.2%

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

      1. unpow2 30.2%

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

      2. sub-neg 30.2%

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

      3. unpow2 30.2%

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

      4. times-frac 31.0%

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

      5. associate-*r/ 31.0%

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

      6. metadata-eval 31.0%

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

      7. distribute-neg-frac 31.0%

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

      8. metadata-eval 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in l around 0 44.5%

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

      1. *-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. unpow2 44.5%

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

      4. times-frac 58.4%

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

      5. distribute-rgt-in 58.4%

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

      6. associate-*r/ 58.4%

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

      7. metadata-eval 58.4%

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

      8. distribute-neg-frac 58.4%

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

      9. metadata-eval 58.4%

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

   9. Simplified 71.8%

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

   if 2.45e207 < l < 2.50000000000000011e244

   1. Initial program 20.3%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in l around inf 41.0%

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

      1. unpow2 41.0%

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

      2. sub-neg 41.0%

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

      3. unpow2 41.0%

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

      4. times-frac 41.0%

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

      5. associate-*r/ 41.0%

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

      6. metadata-eval 41.0%

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

      7. distribute-neg-frac 41.0%

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

      8. metadata-eval 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in l around 0 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. times-frac 41.0%

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

      4. fma-neg 41.0%

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

      5. associate-*r/ 41.0%

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

      6. metadata-eval 41.0%

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

      7. distribute-neg-frac 41.0%

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

      8. metadata-eval 41.0%

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

      9. associate-*r* 61.4%

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

      10. fma-udef 61.4%

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

      11. distribute-rgt-out 41.4%

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

   9. Simplified 61.4%

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

   10. Taylor expanded in U around 0 90.3%

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

   if 2.50000000000000011e244 < l

   1. Initial program 14.0%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in l around inf 28.4%

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

      1. unpow2 28.4%

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

      2. sub-neg 28.4%

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

      3. unpow2 28.4%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 34.7%

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

      10. fma-udef 34.7%

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

      11. distribute-rgt-out 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in l around 0 72.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+207}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 7: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\\ \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot t_1}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- (/ (* n U*) Om) 2.0))))
   (if (<= l 1.05e+131)
     (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
     (if (<= l 3.9e+206)
       (*
        (sqrt 2.0)
        (* l (sqrt (* U (* (/ n Om) (+ -2.0 (* (/ n Om) (- U* U))))))))
       (if (<= l 6.4e+245)
         (sqrt (* 2.0 (* U (* l (/ (* l t_1) Om)))))
         (* (* l (sqrt 2.0)) (sqrt (/ (* U t_1) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (((n * U_42_) / Om) - 2.0);
	double tmp;
	if (l <= 1.05e+131) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 3.9e+206) {
		tmp = sqrt(2.0) * (l * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))));
	} else if (l <= 6.4e+245) {
		tmp = sqrt((2.0 * (U * (l * ((l * t_1) / Om)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((U * t_1) / 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 = n * (((n * u_42) / om) - 2.0d0)
    if (l <= 1.05d+131) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else if (l <= 3.9d+206) then
        tmp = sqrt(2.0d0) * (l * sqrt((u * ((n / om) * ((-2.0d0) + ((n / om) * (u_42 - u)))))))
    else if (l <= 6.4d+245) then
        tmp = sqrt((2.0d0 * (u * (l * ((l * t_1) / om)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((u * t_1) / 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 = n * (((n * U_42_) / Om) - 2.0);
	double tmp;
	if (l <= 1.05e+131) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else if (l <= 3.9e+206) {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))));
	} else if (l <= 6.4e+245) {
		tmp = Math.sqrt((2.0 * (U * (l * ((l * t_1) / Om)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((U * t_1) / Om));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = n * (((n * U_42_) / Om) - 2.0)
	tmp = 0
	if l <= 1.05e+131:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	elif l <= 3.9e+206:
		tmp = math.sqrt(2.0) * (l * math.sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))))
	elif l <= 6.4e+245:
		tmp = math.sqrt((2.0 * (U * (l * ((l * t_1) / Om)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((U * t_1) / Om))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))
	tmp = 0.0
	if (l <= 1.05e+131)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (l <= 3.9e+206)
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(U * Float64(Float64(n / Om) * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U))))))));
	elseif (l <= 6.4e+245)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(l * t_1) / Om)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(U * t_1) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (((n * U_42_) / Om) - 2.0);
	tmp = 0.0;
	if (l <= 1.05e+131)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	elseif (l <= 3.9e+206)
		tmp = sqrt(2.0) * (l * sqrt((U * ((n / Om) * (-2.0 + ((n / Om) * (U_42_ - U)))))));
	elseif (l <= 6.4e+245)
		tmp = sqrt((2.0 * (U * (l * ((l * t_1) / Om)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((U * t_1) / 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[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.05e+131], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.9e+206], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(U * N[(N[(n / Om), $MachinePrecision] * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.4e+245], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(l * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * t$95$1), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.04999999999999993e131

   1. Initial program 50.8%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in Om around inf 45.5%

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

   if 1.04999999999999993e131 < l < 3.9e206

   1. Initial program 29.0%

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

   3. Simplified 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. Step-by-step derivation

      1. sqrt-prod 50.4%

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

   5. Applied egg-rr 50.4%

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

   6. Taylor expanded in l around inf 44.5%

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

      1. *-commutative 44.5%

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

      2. sub-neg 44.5%

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

      3. unpow2 44.5%

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

      4. times-frac 58.5%

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

      5. distribute-rgt-in 58.5%

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

      6. associate-*r/ 58.5%

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

      7. metadata-eval 58.5%

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

      8. distribute-neg-frac 58.5%

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

      9. metadata-eval 58.5%

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

   8. Simplified 71.6%

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

   if 3.9e206 < l < 6.40000000000000048e245

   1. Initial program 20.3%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in l around inf 41.0%

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

      1. unpow2 41.0%

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

      2. sub-neg 41.0%

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

      3. unpow2 41.0%

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

      4. times-frac 41.0%

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

      5. associate-*r/ 41.0%

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

      6. metadata-eval 41.0%

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

      7. distribute-neg-frac 41.0%

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

      8. metadata-eval 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in l around 0 41.0%

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

      1. unpow2 41.0%

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

      2. unpow2 41.0%

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

      3. times-frac 41.0%

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

      4. fma-neg 41.0%

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

      5. associate-*r/ 41.0%

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

      6. metadata-eval 41.0%

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

      7. distribute-neg-frac 41.0%

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

      8. metadata-eval 41.0%

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

      9. associate-*r* 61.4%

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

      10. fma-udef 61.4%

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

      11. distribute-rgt-out 41.4%

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

   9. Simplified 61.4%

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

   10. Taylor expanded in U around 0 90.3%

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

   if 6.40000000000000048e245 < l

   1. Initial program 14.0%

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

   3. Simplified 14.3%

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

   4. Taylor expanded in l around inf 28.4%

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

      1. unpow2 28.4%

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

      2. sub-neg 28.4%

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

      3. unpow2 28.4%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.4%

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

      1. unpow2 28.4%

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

      2. unpow2 28.4%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 34.7%

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

      10. fma-udef 34.7%

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

      11. distribute-rgt-out 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in U around 0 46.9%

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

   11. Taylor expanded in l around 0 72.2%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+206}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{U \cdot \left(\frac{n}{Om} \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right)}\right)\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+245}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 8: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\\ \mathbf{if}\;Om \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t_1}\\ \mathbf{elif}\;Om \leq 5.7 \cdot 10^{+21}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U}{\frac{Om}{n \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (* l (/ l Om)) -2.0 t)))
   (if (<= Om -6.5e-49)
     (sqrt (* (* U (* 2.0 n)) t_1))
     (if (<= Om 5.7e+21)
       (* (* l (sqrt 2.0)) (sqrt (/ U (/ Om (* n (+ -2.0 (/ U* (/ Om n))))))))
       (sqrt (* 2.0 (* U (* n t_1))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((l * (l / Om)), -2.0, t);
	double tmp;
	if (Om <= -6.5e-49) {
		tmp = sqrt(((U * (2.0 * n)) * t_1));
	} else if (Om <= 5.7e+21) {
		tmp = (l * sqrt(2.0)) * sqrt((U / (Om / (n * (-2.0 + (U_42_ / (Om / n)))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t_1))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(l * Float64(l / Om)), -2.0, t)
	tmp = 0.0
	if (Om <= -6.5e-49)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t_1));
	elseif (Om <= 5.7e+21)
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(U / Float64(Om / Float64(n * Float64(-2.0 + Float64(U_42_ / Float64(Om / n))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_1))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[Om, -6.5e-49], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 5.7e+21], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U / N[(Om / N[(n * N[(-2.0 + N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -6.49999999999999968e-49

   1. Initial program 55.1%

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

      1. associate-*l/ 66.3%

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

   3. Applied egg-rr 66.3%

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

   4. Taylor expanded in Om around inf 52.9%

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

      1. +-commutative 52.9%

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

      2. *-commutative 52.9%

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

      3. unpow2 52.9%

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

      4. associate-*r/ 62.1%

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

      5. fma-udef 62.1%

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

   6. Simplified 62.1%

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

   if -6.49999999999999968e-49 < Om < 5.7e21

   1. Initial program 33.9%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in l around inf 34.7%

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

      1. unpow2 34.7%

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

      2. sub-neg 34.7%

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

      3. unpow2 34.7%

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

      4. times-frac 35.9%

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

      5. associate-*r/ 35.9%

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

      6. metadata-eval 35.9%

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

      7. distribute-neg-frac 35.9%

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

      8. metadata-eval 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in l around 0 34.7%

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

      1. unpow2 34.7%

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

      2. unpow2 34.7%

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

      3. times-frac 35.9%

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

      4. fma-neg 35.9%

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

      5. associate-*r/ 35.9%

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

      6. metadata-eval 35.9%

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

      7. distribute-neg-frac 35.9%

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

      8. metadata-eval 35.9%

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

      9. associate-*r* 38.2%

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

      10. fma-udef 38.2%

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

      11. distribute-rgt-out 32.0%

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

   9. Simplified 46.1%

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

   10. Taylor expanded in U around 0 46.3%

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

   11. Taylor expanded in l around 0 25.2%

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

       1. *-commutative 25.2%

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

       2. associate-/l* 32.1%

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

       3. sub-neg 32.1%

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

       4. *-commutative 32.1%

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

       5. metadata-eval 32.1%

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

       6. +-commutative 32.1%

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

       7. *-commutative 32.1%

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

       8. associate-/l* 32.1%

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

   13. Simplified 32.1%

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

   if 5.7e21 < Om

   1. Initial program 52.2%

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

      1. associate-*l/ 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in n around 0 53.3%

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

      1. cancel-sign-sub-inv 53.3%

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

      2. metadata-eval 53.3%

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

      3. +-commutative 53.3%

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

      4. *-commutative 53.3%

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

      5. unpow2 53.3%

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

      6. associate-*r/ 60.1%

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

      7. fma-udef 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)}\\ \mathbf{elif}\;Om \leq 5.7 \cdot 10^{+21}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U}{\frac{Om}{n \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \end{array} \]

Alternative 9: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.5e+129)
   (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
   (* (* l (sqrt 2.0)) (sqrt (/ (* U (* n (- (/ (* n 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 <= 7.5e+129) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / 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 <= 7.5d+129) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((u * (n * (((n * u_42) / 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 <= 7.5e+129) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7.5e+129:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.5e+129)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 7.5e+129)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / 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, 7.5e+129], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.4999999999999998e129

   1. Initial program 50.8%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in Om around inf 45.5%

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

   if 7.4999999999999998e129 < l

   1. Initial program 20.8%

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

   3. Simplified 31.1%

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

   4. Taylor expanded in l around inf 32.2%

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

      1. unpow2 32.2%

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

      2. sub-neg 32.2%

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

      3. unpow2 32.2%

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

      4. times-frac 32.5%

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

      5. associate-*r/ 32.5%

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

      6. metadata-eval 32.5%

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

      7. distribute-neg-frac 32.5%

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

      8. metadata-eval 32.5%

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

   6. Simplified 32.5%

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

   7. Taylor expanded in l around 0 32.2%

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

      1. unpow2 32.2%

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

      2. unpow2 32.2%

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

      3. times-frac 32.5%

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

      4. fma-neg 32.5%

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

      5. associate-*r/ 32.5%

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

      6. metadata-eval 32.5%

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

      7. distribute-neg-frac 32.5%

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

      8. metadata-eval 32.5%

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

      9. associate-*r* 44.9%

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

      10. fma-udef 44.9%

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

      11. distribute-rgt-out 39.9%

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

   9. Simplified 45.0%

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

   10. Taylor expanded in U around 0 59.5%

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

   11. Taylor expanded in l around 0 63.2%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 10: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.05e+154)
   (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
   (* (* l (sqrt 2.0)) (sqrt (/ (* U (* n (- (/ (* n U*) Om) 2.0))) Om)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.05e+154) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((U * (n * (((n * U_42_) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.05e+154)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(U * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.05e+154], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.04999999999999997e154

   1. Initial program 51.0%

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

      1. associate-*l/ 55.4%

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

   3. Applied egg-rr 55.4%

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

   4. Taylor expanded in n around 0 45.9%

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

      1. cancel-sign-sub-inv 45.9%

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

      2. metadata-eval 45.9%

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

      3. +-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. unpow2 45.9%

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

      6. associate-*r/ 49.4%

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

      7. fma-udef 49.4%

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

   6. Simplified 49.4%

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

   if 1.04999999999999997e154 < l

   1. Initial program 15.3%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in l around inf 28.1%

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

      1. unpow2 28.1%

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

      2. sub-neg 28.1%

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

      3. unpow2 28.1%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.1%

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

      1. unpow2 28.1%

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

      2. unpow2 28.1%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

      9. associate-*r* 42.7%

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

      10. fma-udef 42.7%

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

      11. distribute-rgt-out 37.0%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in U around 0 59.0%

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

   11. Taylor expanded in l around 0 65.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{U \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 11: 54.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e+119)
   (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
   (sqrt
    (* 2.0 (* U (* l (* (/ 1.0 Om) (* (* n l) (+ -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 tmp;
	if (l <= 4.5e+119) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else {
		tmp = sqrt((2.0 * (U * (l * ((1.0 / Om) * ((n * l) * (-2.0 + (U_42_ / (Om / n)))))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.5e+119)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(1.0 / Om) * Float64(Float64(n * l) * Float64(-2.0 + Float64(U_42_ / Float64(Om / n)))))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.5e+119], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(1.0 / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(-2.0 + N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.5000000000000002e119

   1. Initial program 51.0%

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

      1. associate-*l/ 55.6%

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

   3. Applied egg-rr 55.6%

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

   4. Taylor expanded in n around 0 46.2%

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

      1. cancel-sign-sub-inv 46.2%

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

      2. metadata-eval 46.2%

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

      3. +-commutative 46.2%

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

      4. *-commutative 46.2%

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

      5. unpow2 46.2%

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

      6. associate-*r/ 49.8%

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

      7. fma-udef 49.8%

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

   6. Simplified 49.8%

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

   if 4.5000000000000002e119 < l

   1. Initial program 20.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in l around inf 33.9%

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

      1. unpow2 33.9%

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

      2. sub-neg 33.9%

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

      3. unpow2 33.9%

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

      4. times-frac 34.1%

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

      5. associate-*r/ 34.1%

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

      6. metadata-eval 34.1%

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

      7. distribute-neg-frac 34.1%

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

      8. metadata-eval 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in l around 0 33.9%

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

      1. unpow2 33.9%

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

      2. unpow2 33.9%

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

      3. times-frac 34.1%

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

      4. fma-neg 34.1%

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

      5. associate-*r/ 34.1%

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

      6. metadata-eval 34.1%

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

      7. distribute-neg-frac 34.1%

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

      8. metadata-eval 34.1%

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

      9. associate-*r* 46.3%

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

      10. fma-udef 46.3%

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

      11. distribute-rgt-out 41.4%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in U around 0 60.4%

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

       1. div-inv 60.4%

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

       2. sub-neg 60.4%

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

       3. *-commutative 60.4%

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

       4. metadata-eval 60.4%

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

   12. Applied egg-rr 60.4%

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

       1. *-commutative 60.4%

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

       2. associate-*r* 60.4%

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

       3. +-commutative 60.4%

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

       4. *-commutative 60.4%

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

       5. associate-/l* 60.4%

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

   14. Simplified 60.4%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.45e+119)
   (sqrt (* (- t (* 2.0 (/ (* l l) Om))) (* 2.0 (* n U))))
   (sqrt
    (* 2.0 (* U (* l (* (/ 1.0 Om) (* (* n l) (+ -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 tmp;
	if (l <= 1.45e+119) {
		tmp = sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (l * ((1.0 / Om) * ((n * l) * (-2.0 + (U_42_ / (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) :: tmp
    if (l <= 1.45d+119) then
        tmp = sqrt(((t - (2.0d0 * ((l * l) / om))) * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (l * ((1.0d0 / om) * ((n * l) * ((-2.0d0) + (u_42 / (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 tmp;
	if (l <= 1.45e+119) {
		tmp = Math.sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (l * ((1.0 / Om) * ((n * l) * (-2.0 + (U_42_ / (Om / n)))))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.45e+119:
		tmp = math.sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (l * ((1.0 / Om) * ((n * l) * (-2.0 + (U_42_ / (Om / n)))))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.45e+119)
		tmp = sqrt(Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(1.0 / Om) * Float64(Float64(n * l) * Float64(-2.0 + Float64(U_42_ / Float64(Om / n)))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.45e+119)
		tmp = sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (l * ((1.0 / Om) * ((n * l) * (-2.0 + (U_42_ / (Om / n)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.45e+119], N[Sqrt[N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(1.0 / Om), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(-2.0 + N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.45000000000000004e119

   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 51.2%

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

   4. Taylor expanded in Om around inf 45.0%

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

      1. unpow2 45.0%

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

   6. Simplified 45.0%

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

   if 1.45000000000000004e119 < l

   1. Initial program 20.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in l around inf 33.9%

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

      1. unpow2 33.9%

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

      2. sub-neg 33.9%

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

      3. unpow2 33.9%

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

      4. times-frac 34.1%

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

      5. associate-*r/ 34.1%

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

      6. metadata-eval 34.1%

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

      7. distribute-neg-frac 34.1%

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

      8. metadata-eval 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in l around 0 33.9%

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

      1. unpow2 33.9%

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

      2. unpow2 33.9%

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

      3. times-frac 34.1%

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

      4. fma-neg 34.1%

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

      5. associate-*r/ 34.1%

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

      6. metadata-eval 34.1%

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

      7. distribute-neg-frac 34.1%

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

      8. metadata-eval 34.1%

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

      9. associate-*r* 46.3%

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

      10. fma-udef 46.3%

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

      11. distribute-rgt-out 41.4%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in U around 0 60.4%

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

       1. div-inv 60.4%

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

       2. sub-neg 60.4%

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

       3. *-commutative 60.4%

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

       4. metadata-eval 60.4%

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

   12. Applied egg-rr 60.4%

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

       1. *-commutative 60.4%

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

       2. associate-*r* 60.4%

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

       3. +-commutative 60.4%

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

       4. *-commutative 60.4%

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

       5. associate-/l* 60.4%

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

   14. Simplified 60.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\frac{1}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \left(-2 + \frac{U*}{\frac{Om}{n}}\right)\right)\right)\right)\right)}\\ \end{array} \]

Alternative 13: 53.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e+119)
   (sqrt (* (- t (* 2.0 (/ (* l l) Om))) (* 2.0 (* n U))))
   (sqrt (* 2.0 (* U (* l (/ (* l (* n (- (/ (* n U*) Om) 2.0))) Om)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+119) {
		tmp = sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.15d+119) then
        tmp = sqrt(((t - (2.0d0 * ((l * l) / om))) * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (l * ((l * (n * (((n * u_42) / om) - 2.0d0))) / om)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+119) {
		tmp = Math.sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e+119:
		tmp = math.sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e+119)
		tmp = sqrt(Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(l * Float64(n * Float64(Float64(Float64(n * U_42_) / Om) - 2.0))) / Om)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e+119)
		tmp = sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (l * ((l * (n * (((n * U_42_) / Om) - 2.0))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e+119], N[Sqrt[N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(l * N[(n * N[(N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e119

   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 51.2%

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

   4. Taylor expanded in Om around inf 45.0%

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

      1. unpow2 45.0%

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

   6. Simplified 45.0%

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

   if 1.15e119 < l

   1. Initial program 20.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in l around inf 33.9%

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

      1. unpow2 33.9%

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

      2. sub-neg 33.9%

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

      3. unpow2 33.9%

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

      4. times-frac 34.1%

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

      5. associate-*r/ 34.1%

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

      6. metadata-eval 34.1%

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

      7. distribute-neg-frac 34.1%

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

      8. metadata-eval 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in l around 0 33.9%

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

      1. unpow2 33.9%

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

      2. unpow2 33.9%

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

      3. times-frac 34.1%

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

      4. fma-neg 34.1%

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

      5. associate-*r/ 34.1%

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

      6. metadata-eval 34.1%

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

      7. distribute-neg-frac 34.1%

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

      8. metadata-eval 34.1%

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

      9. associate-*r* 46.3%

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

      10. fma-udef 46.3%

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

      11. distribute-rgt-out 41.4%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in U around 0 60.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot \left(n \cdot \left(\frac{n \cdot U*}{Om} - 2\right)\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 14: 34.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -9.5 \cdot 10^{+73}:\\ \;\;\;\;{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 3.5 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -9.5e+73)
   (pow (* (* U (* 2.0 n)) t) 0.5)
   (if (<= Om 3.5e-226)
     (sqrt (* 2.0 (* -2.0 (/ U (/ Om (* n (* l l)))))))
     (sqrt (* (* 2.0 n) (* U t))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -9.5e+73) {
		tmp = pow(((U * (2.0 * n)) * t), 0.5);
	} else if (Om <= 3.5e-226) {
		tmp = sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-9.5d+73)) then
        tmp = ((u * (2.0d0 * n)) * t) ** 0.5d0
    else if (om <= 3.5d-226) then
        tmp = sqrt((2.0d0 * ((-2.0d0) * (u / (om / (n * (l * l)))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -9.5e+73) {
		tmp = Math.pow(((U * (2.0 * n)) * t), 0.5);
	} else if (Om <= 3.5e-226) {
		tmp = Math.sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -9.5e+73:
		tmp = math.pow(((U * (2.0 * n)) * t), 0.5)
	elif Om <= 3.5e-226:
		tmp = math.sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -9.5e+73)
		tmp = Float64(Float64(U * Float64(2.0 * n)) * t) ^ 0.5;
	elseif (Om <= 3.5e-226)
		tmp = sqrt(Float64(2.0 * Float64(-2.0 * Float64(U / Float64(Om / Float64(n * Float64(l * l)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -9.5e+73)
		tmp = ((U * (2.0 * n)) * t) ^ 0.5;
	elseif (Om <= 3.5e-226)
		tmp = sqrt((2.0 * (-2.0 * (U / (Om / (n * (l * l)))))));
	else
		tmp = sqrt(((2.0 * n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -9.5e+73], N[Power[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[Om, 3.5e-226], N[Sqrt[N[(2.0 * N[(-2.0 * N[(U / N[(Om / N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -9.4999999999999996e73

   1. Initial program 53.9%

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

      1. associate-*l/ 67.9%

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

   3. Applied egg-rr 67.9%

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

   4. Taylor expanded in t around inf 44.8%

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

      1. pow1/2 46.4%

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

      2. *-commutative 46.4%

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

      3. *-commutative 46.4%

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

      4. *-commutative 46.4%

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

   6. Applied egg-rr 46.4%

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

   if -9.4999999999999996e73 < Om < 3.5e-226

   1. Initial program 41.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in l around inf 41.5%

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

      1. unpow2 41.5%

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

      2. sub-neg 41.5%

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

      3. unpow2 41.5%

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

      4. times-frac 44.4%

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

      5. associate-*r/ 44.4%

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

      6. metadata-eval 44.4%

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

      7. distribute-neg-frac 44.4%

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

      8. metadata-eval 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in n around 0 31.6%

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

      1. associate-/l* 34.1%

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

      2. *-commutative 34.1%

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

      3. unpow2 34.1%

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

   9. Simplified 34.1%

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

   if 3.5e-226 < Om

   1. Initial program 44.6%

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

      1. associate-*l/ 48.9%

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

   3. Applied egg-rr 48.9%

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

   4. Taylor expanded in t around inf 36.2%

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

      1. *-un-lft-identity 36.2%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      2. *-commutative 36.2%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      3. *-commutative 36.2%

         \[\leadsto 1 \cdot \sqrt{t \cdot \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)}} \]

      4. *-commutative 36.2%

         \[\leadsto 1 \cdot \sqrt{t \cdot \left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right)} \]

   6. Applied egg-rr 36.2%

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

      1. *-lft-identity 36.2%

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

      2. associate-*r* 41.6%

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

   8. Simplified 41.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.5 \cdot 10^{+73}:\\ \;\;\;\;{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 3.5 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U}{\frac{Om}{n \cdot \left(\ell \cdot \ell\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 15: 48.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{-2 \cdot \left(n \cdot \ell\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.05e+154)
   (sqrt (* (- t (* 2.0 (/ (* l l) Om))) (* 2.0 (* n U))))
   (sqrt (* 2.0 (* U (* l (/ (* -2.0 (* n 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 <= 1.05e+154) {
		tmp = sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (l * ((-2.0 * (n * 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 <= 1.05d+154) then
        tmp = sqrt(((t - (2.0d0 * ((l * l) / om))) * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (l * (((-2.0d0) * (n * 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 <= 1.05e+154) {
		tmp = Math.sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (l * ((-2.0 * (n * l)) / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.05e+154:
		tmp = math.sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (l * ((-2.0 * (n * l)) / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.05e+154)
		tmp = sqrt(Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(-2.0 * Float64(n * 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 <= 1.05e+154)
		tmp = sqrt(((t - (2.0 * ((l * l) / Om))) * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (l * ((-2.0 * (n * 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, 1.05e+154], N[Sqrt[N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(-2.0 * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.04999999999999997e154

   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 51.2%

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

   4. Taylor expanded in Om around inf 44.7%

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

      1. unpow2 44.7%

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

   6. Simplified 44.7%

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

   if 1.04999999999999997e154 < l

   1. Initial program 15.3%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in l around inf 28.1%

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

      1. unpow2 28.1%

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

      2. sub-neg 28.1%

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

      3. unpow2 28.1%

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

      4. times-frac 28.4%

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

      5. associate-*r/ 28.4%

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

      6. metadata-eval 28.4%

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

      7. distribute-neg-frac 28.4%

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

      8. metadata-eval 28.4%

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

   6. Simplified 28.4%

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

   7. Taylor expanded in l around 0 28.1%

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

      1. unpow2 28.1%

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

      2. unpow2 28.1%

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

      3. times-frac 28.4%

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

      4. fma-neg 28.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -2 \cdot \frac{1}{Om}\right)}\right)\right)\right)} \]

      5. associate-*r/ 28.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      6. metadata-eval 28.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

      7. distribute-neg-frac 28.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \color{blue}{\frac{-2}{Om}}\right)\right)\right)\right)} \]

      8. metadata-eval 28.4%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

      9. associate-*r* 42.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)\right)}\right)} \]

      10. fma-udef 42.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(n \cdot \color{blue}{\left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)}\right)\right)\right)\right)} \]

      11. distribute-rgt-out 37.0%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot n + \frac{-2}{Om} \cdot n\right)}\right)\right)\right)} \]

   9. Simplified 42.7%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + -2\right)\right)\right)\right)}\right)} \]

   10. Taylor expanded in n around 0 47.1%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(-2 \cdot \frac{\ell \cdot n}{Om}\right)}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 47.1%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{-2 \cdot \left(\ell \cdot n\right)}{Om}}\right)\right)} \]

   12. Simplified 47.1%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{-2 \cdot \left(\ell \cdot n\right)}{Om}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{-2 \cdot \left(n \cdot \ell\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 16: 43.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{+67}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot -2}{\frac{Om}{n}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.35e+67)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* 2.0 (* U (* l (/ (* l -2.0) (/ Om n))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.35e+67) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (l * ((l * -2.0) / (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) :: tmp
    if (l <= 1.35d+67) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (l * ((l * (-2.0d0)) / (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 tmp;
	if (l <= 1.35e+67) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (l * ((l * -2.0) / (Om / n))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.35e+67:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (l * ((l * -2.0) / (Om / n))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.35e+67)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(l * -2.0) / Float64(Om / n))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.35e+67)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (l * ((l * -2.0) / (Om / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.35e+67], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(l * -2.0), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.35e67

   1. Initial program 50.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around 0 39.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 40.9%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   6. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   if 1.35e67 < l

   1. Initial program 30.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around inf 35.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. sub-neg 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2 \cdot \frac{1}{Om}\right)\right)}\right)\right)\right)} \]

      3. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)} \]

      4. times-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)} \]

      5. associate-*r/ 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)\right)} \]

      6. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)\right)} \]

      7. distribute-neg-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \color{blue}{\frac{-2}{Om}}\right)\right)\right)\right)} \]

      8. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

   6. Simplified 37.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}} \]

   7. Taylor expanded in l around 0 35.5%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      3. times-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      4. fma-neg 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -2 \cdot \frac{1}{Om}\right)}\right)\right)\right)} \]

      5. associate-*r/ 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      6. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

      7. distribute-neg-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \color{blue}{\frac{-2}{Om}}\right)\right)\right)\right)} \]

      8. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

      9. associate-*r* 47.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)\right)}\right)} \]

      10. fma-udef 47.2%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(n \cdot \color{blue}{\left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)}\right)\right)\right)\right)} \]

      11. distribute-rgt-out 43.4%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot n + \frac{-2}{Om} \cdot n\right)}\right)\right)\right)} \]

   9. Simplified 47.2%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + -2\right)\right)\right)\right)}\right)} \]

   10. Taylor expanded in U around 0 58.6%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot \left(n \cdot \left(\frac{U* \cdot n}{Om} - 2\right)\right)}{Om}}\right)\right)} \]

   11. Taylor expanded in n around 0 48.4%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(-2 \cdot \frac{\ell \cdot n}{Om}\right)}\right)\right)} \]
   12. Step-by-step derivation

       1. associate-/l* 34.0%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{n}}}\right)\right)\right)} \]

       2. associate-*r/ 34.0%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{-2 \cdot \ell}{\frac{Om}{n}}}\right)\right)} \]

   13. Simplified 34.0%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{-2 \cdot \ell}{\frac{Om}{n}}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{+67}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell \cdot -2}{\frac{Om}{n}}\right)\right)}\\ \end{array} \]

Alternative 17: 46.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{+67}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{-2 \cdot \left(n \cdot \ell\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.8e+67)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* 2.0 (* U (* l (/ (* -2.0 (* n 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 <= 4.8e+67) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (l * ((-2.0 * (n * 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 <= 4.8d+67) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (l * (((-2.0d0) * (n * 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 <= 4.8e+67) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (l * ((-2.0 * (n * l)) / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.8e+67:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (l * ((-2.0 * (n * l)) / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.8e+67)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(l * Float64(Float64(-2.0 * Float64(n * 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 <= 4.8e+67)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (l * ((-2.0 * (n * 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, 4.8e+67], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(l * N[(N[(-2.0 * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.80000000000000004e67

   1. Initial program 50.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around 0 39.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 40.9%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   6. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   if 4.80000000000000004e67 < l

   1. Initial program 30.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around inf 35.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. sub-neg 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + \left(-2 \cdot \frac{1}{Om}\right)\right)}\right)\right)\right)} \]

      3. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)} \]

      4. times-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} + \left(-2 \cdot \frac{1}{Om}\right)\right)\right)\right)\right)} \]

      5. associate-*r/ 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)\right)} \]

      6. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \left(-\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)\right)} \]

      7. distribute-neg-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \color{blue}{\frac{-2}{Om}}\right)\right)\right)\right)} \]

      8. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

   6. Simplified 37.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)\right)\right)}} \]

   7. Taylor expanded in l around 0 35.5%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. unpow2 35.5%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      3. times-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U* - U}{Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      4. fma-neg 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -2 \cdot \frac{1}{Om}\right)}\right)\right)\right)} \]

      5. associate-*r/ 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      6. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, -\frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

      7. distribute-neg-frac 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \color{blue}{\frac{-2}{Om}}\right)\right)\right)\right)} \]

      8. metadata-eval 37.7%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{\color{blue}{-2}}{Om}\right)\right)\right)\right)} \]

      9. associate-*r* 47.2%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(n \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)\right)\right)\right)}\right)} \]

      10. fma-udef 47.2%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \left(n \cdot \color{blue}{\left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)}\right)\right)\right)\right)} \]

      11. distribute-rgt-out 43.4%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\left(\frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot n + \frac{-2}{Om} \cdot n\right)}\right)\right)\right)} \]

   9. Simplified 47.2%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{n}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n}{Om} + -2\right)\right)\right)\right)}\right)} \]

   10. Taylor expanded in n around 0 48.4%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(-2 \cdot \frac{\ell \cdot n}{Om}\right)}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 48.4%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{-2 \cdot \left(\ell \cdot n\right)}{Om}}\right)\right)} \]

   12. Simplified 48.4%

       \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\frac{-2 \cdot \left(\ell \cdot n\right)}{Om}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{+67}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\ell \cdot \frac{-2 \cdot \left(n \cdot \ell\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 18: 41.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+69}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.7e+69)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* -4.0 (/ (* U (* n (* 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 <= 1.7e+69) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((-4.0 * ((U * (n * (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 <= 1.7d+69) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt(((-4.0d0) * ((u * (n * (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 <= 1.7e+69) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((-4.0 * ((U * (n * (l * l))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.7e+69:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((-4.0 * ((U * (n * (l * l))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.7e+69)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(n * Float64(l * l))) / Om)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.7e+69)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((-4.0 * ((U * (n * (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, 1.7e+69], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.69999999999999993e69

   1. Initial program 50.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around 0 39.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 40.9%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   6. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   if 1.69999999999999993e69 < l

   1. Initial program 30.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 40.3%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 40.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in Om around inf 27.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 27.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. *-commutative 27.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right)} \]

      3. unpow2 27.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot -2 + t\right)} \]

      4. associate-*r/ 35.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot -2 + t\right)} \]

      5. fma-udef 35.8%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)}} \]

   6. Simplified 35.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)}} \]

   7. Taylor expanded in l around inf 32.2%

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   8. Step-by-step derivation

      1. unpow2 32.2%

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{Om}} \]

   9. Simplified 32.2%

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+69}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]

Alternative 19: 37.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-71}:\\ \;\;\;\;{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 2e-71)
   (pow (* (* U (* 2.0 n)) t) 0.5)
   (sqrt (* (* 2.0 n) (* U t)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 2e-71) {
		tmp = pow(((U * (2.0 * n)) * t), 0.5);
	} else {
		tmp = sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 2d-71) then
        tmp = ((u * (2.0d0 * n)) * t) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 2e-71) {
		tmp = Math.pow(((U * (2.0 * n)) * t), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 2e-71:
		tmp = math.pow(((U * (2.0 * n)) * t), 0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 2e-71)
		tmp = Float64(Float64(U * Float64(2.0 * n)) * t) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= 2e-71)
		tmp = ((U * (2.0 * n)) * t) ^ 0.5;
	else
		tmp = sqrt(((2.0 * n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 2e-71], N[Power[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.9999999999999998e-71

   1. Initial program 45.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 51.0%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 51.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in t around inf 27.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
   5. Step-by-step derivation

      1. pow1/2 30.4%

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}} \]

      2. *-commutative 30.4%

         \[\leadsto {\color{blue}{\left(t \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)}}^{0.5} \]

      3. *-commutative 30.4%

         \[\leadsto {\left(t \cdot \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)}\right)}^{0.5} \]

      4. *-commutative 30.4%

         \[\leadsto {\left(t \cdot \left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right)\right)}^{0.5} \]

   6. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{{\left(t \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}} \]

   if 1.9999999999999998e-71 < Om

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 53.5%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 53.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in t around inf 41.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 41.5%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      2. *-commutative 41.5%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      3. *-commutative 41.5%

         \[\leadsto 1 \cdot \sqrt{t \cdot \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)}} \]

      4. *-commutative 41.5%

         \[\leadsto 1 \cdot \sqrt{t \cdot \left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right)} \]

   6. Applied egg-rr 41.5%

      \[\leadsto \color{blue}{1 \cdot \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}} \]
   7. Step-by-step derivation

      1. *-lft-identity 41.5%

         \[\leadsto \color{blue}{\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}} \]

      2. associate-*r* 47.5%

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   8. Simplified 47.5%

      \[\leadsto \color{blue}{\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-71}:\\ \;\;\;\;{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 20: 38.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* U (* n t))) 0.5))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (U * (n * t))), 0.5);
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (2.0d0 * (u * (n * t))) ** 0.5d0
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((2.0 * (U * (n * t))), 0.5);
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (U * (n * t))), 0.5)

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5
end

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * (U * (n * t))) ^ 0.5;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 46.1%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.3%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

4. Taylor expanded in l around 0 33.3%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. pow1/2 34.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

6. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

7. Final simplification 34.6%

   \[\leadsto {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \]

Alternative 21: 36.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 46.1%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.3%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

4. Taylor expanded in l around 0 33.3%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

5. Final simplification 33.3%

   \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Reproduce

herbie shell --seed 2023283 
(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*))))))