Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 66.6%

Time: 50.8s

Alternatives: 19

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(t_1, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* U (* 2.0 n))
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 0.0)
     (sqrt (* 2.0 (* n (* U (fma t_1 (- U* U) (fma -2.0 (* l (/ l Om)) t))))))
     (if (<= t_2 1e+144)
       t_2
       (*
        (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * (n * (U * fma(t_1, (U_42_ - U), fma(-2.0, (l * (l / Om)), t))))));
	} else if (t_2 <= 1e+144) {
		tmp = t_2;
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * fma(t_1, Float64(U_42_ - U), fma(-2.0, Float64(l * Float64(l / Om)), t))))));
	elseif (t_2 <= 1e+144)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision] + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+144], t$95$2, N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $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 13.7%

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

   3. Simplified 13.2%

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

      1. pow1 13.2%

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

   5. Applied egg-rr 13.2%

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

      1. unpow1 13.2%

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

      2. associate-*l* 30.2%

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

      3. fma-def 30.2%

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

      4. associate-*r* 39.9%

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

      5. fma-udef 39.9%

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

      6. *-commutative 39.9%

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

      7. associate-*r/ 39.9%

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

      8. unpow2 39.9%

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

      9. +-commutative 39.9%

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

      10. fma-def 40.0%

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

      11. +-commutative 40.0%

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

      12. unpow2 40.0%

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

      13. associate-*r/ 40.0%

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

      14. fma-def 40.0%

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

   7. Simplified 40.0%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000002e144

   1. Initial program 96.5%

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

   if 1.00000000000000002e144 < (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 20.1%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in l around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. unpow2 22.6%

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

      3. times-frac 26.4%

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

      4. associate-*r/ 26.4%

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

      5. metadata-eval 26.4%

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

      6. distribute-neg-frac 26.4%

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

      7. metadata-eval 26.4%

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

   6. Simplified 26.4%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 2: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t_1 \leq 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* U (* 2.0 n))
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (if (<= t_1 1e+144)
       t_1
       (*
        (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (t_1 <= 1e+144) {
		tmp = t_1;
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((u * (2.0d0 * n)) * ((t - (2.0d0 * ((l * l) / om))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    else if (t_1 <= 1d+144) then
        tmp = t_1
    else
        tmp = sqrt((u * (n * (((n / om) * ((u_42 - u) / om)) + ((-2.0d0) / om))))) * (l * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	} else if (t_1 <= 1e+144) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	elif t_1 <= 1e+144:
		tmp = t_1
	else:
		tmp = math.sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * math.sqrt(2.0))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (t_1 <= 1e+144)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	elseif (t_1 <= 1e+144)
		tmp = t_1;
	else
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+144], t$95$1, N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $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 13.7%

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

   3. Simplified 13.2%

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

   4. Taylor expanded in l around 0 28.1%

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

      1. associate-*r* 13.7%

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

      2. *-commutative 13.7%

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

   6. Simplified 13.7%

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

      1. *-un-lft-identity 13.7%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*l* 28.1%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   8. Applied egg-rr 28.1%

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

      1. *-lft-identity 28.1%

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

      2. associate-*r* 28.1%

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

   10. Simplified 28.1%

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

       1. sqrt-prod 37.6%

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

       2. *-commutative 37.6%

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

   12. Applied egg-rr 37.6%

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

   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*))))) < 1.00000000000000002e144

   1. Initial program 96.5%

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

   if 1.00000000000000002e144 < (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 20.1%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in l around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. unpow2 22.6%

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

      3. times-frac 26.4%

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

      4. associate-*r/ 26.4%

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

      5. metadata-eval 26.4%

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

      6. distribute-neg-frac 26.4%

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

      7. metadata-eval 26.4%

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

   6. Simplified 26.4%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 3: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_1\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - t_1\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U U*)))
        (t_2 (sqrt (* (* U (* 2.0 n)) (- (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_2 0.0)
     (sqrt (* 2.0 (* n (* U (- (fma -2.0 (* l (/ l Om)) t) t_1)))))
     (if (<= t_2 1e+144)
       t_2
       (*
        (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)))))
        (* l (sqrt 2.0)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U - U_42_);
	double t_2 = sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) - t_1)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * (n * (U * (fma(-2.0, (l * (l / Om)), t) - t_1)))));
	} else if (t_2 <= 1e+144) {
		tmp = t_2;
	} else {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))
	t_2 = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - t_1)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - t_1)))));
	elseif (t_2 <= 1e+144)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+144], t$95$2, N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $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 13.7%

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

   3. Simplified 13.2%

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

      1. pow1 13.2%

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

   5. Applied egg-rr 13.2%

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

      1. unpow1 13.2%

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

      2. associate-*l* 30.2%

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

      3. fma-def 30.2%

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

      4. associate-*r* 39.9%

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

      5. fma-udef 39.9%

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

      6. *-commutative 39.9%

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

      7. associate-*r/ 39.9%

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

      8. unpow2 39.9%

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

      9. +-commutative 39.9%

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

      10. fma-def 40.0%

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

      11. +-commutative 40.0%

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

      12. unpow2 40.0%

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

      13. associate-*r/ 40.0%

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

      14. fma-def 40.0%

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

   7. Simplified 40.0%

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

      1. fma-udef 39.9%

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

   9. Applied egg-rr 39.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.00000000000000002e144

   1. Initial program 96.5%

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

   if 1.00000000000000002e144 < (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 20.1%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in l around inf 22.6%

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

      1. sub-neg 22.6%

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

      2. unpow2 22.6%

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

      3. times-frac 26.4%

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

      4. associate-*r/ 26.4%

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

      5. metadata-eval 26.4%

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

      6. distribute-neg-frac 26.4%

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

      7. metadata-eval 26.4%

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

   6. Simplified 26.4%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 4: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{U \cdot U*}{Om} \cdot \left(n \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.95 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+151} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+239}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))))
   (if (<= l 6.2e-80)
     (sqrt (* 2.0 (* t (* n U))))
     (if (<= l 3.8e-43)
       (sqrt (* 2.0 (* n (* (/ (* U U*) Om) (* n (/ (* l l) Om))))))
       (if (<= l 2.8e-37)
         (* (sqrt (* 2.0 n)) (sqrt (* U t)))
         (if (<= l 4.3e-26)
           (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (* l l) (* n (* n (- U U*))))))
           (if (<= l 8.8e+56)
             t_1
             (if (<= l 3.95e+139)
               (sqrt
                (*
                 2.0
                 (*
                  n
                  (*
                   U
                   (+
                    t
                    (*
                     (* l l)
                     (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))
               (if (or (<= l 2.8e+151) (not (<= l 2.7e+239)))
                 (*
                  (sqrt 2.0)
                  (*
                   l
                   (sqrt
                    (* (* n U) (- (/ n (/ (* Om Om) (- U* U))) (/ 2.0 Om))))))
                 t_1)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	double tmp;
	if (l <= 6.2e-80) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else if (l <= 3.8e-43) {
		tmp = sqrt((2.0 * (n * (((U * U_42_) / Om) * (n * ((l * l) / Om))))));
	} else if (l <= 2.8e-37) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (l <= 4.3e-26) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((l * l) * (n * (n * (U - U_42_))))));
	} else if (l <= 8.8e+56) {
		tmp = t_1;
	} else if (l <= 3.95e+139) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	} else if ((l <= 2.8e+151) || !(l <= 2.7e+239)) {
		tmp = sqrt(2.0) * (l * sqrt(((n * U) * ((n / ((Om * Om) / (U_42_ - U))) - (2.0 / Om)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))))
	tmp = 0.0
	if (l <= 6.2e-80)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	elseif (l <= 3.8e-43)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(Float64(U * U_42_) / Om) * Float64(n * Float64(Float64(l * l) / Om))))));
	elseif (l <= 2.8e-37)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (l <= 4.3e-26)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(l * l) * Float64(n * Float64(n * Float64(U - U_42_))))));
	elseif (l <= 8.8e+56)
		tmp = t_1;
	elseif (l <= 3.95e+139)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))));
	elseif ((l <= 2.8e+151) || !(l <= 2.7e+239))
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(n * U) * Float64(Float64(n / Float64(Float64(Om * Om) / Float64(U_42_ - U))) - Float64(2.0 / Om))))));
	else
		tmp = 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[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 6.2e-80], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.8e-43], N[Sqrt[N[(2.0 * N[(n * N[(N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.8e-37], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.3e-26], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(n * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.8e+56], t$95$1, If[LessEqual[l, 3.95e+139], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 2.8e+151], N[Not[LessEqual[l, 2.7e+239]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < 6.20000000000000032e-80

   1. Initial program 49.7%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in l around 0 40.3%

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

      1. associate-*r* 41.4%

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

      2. *-commutative 41.4%

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

   6. Simplified 41.4%

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

   if 6.20000000000000032e-80 < l < 3.7999999999999997e-43

   1. Initial program 65.1%

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

   3. Simplified 56.5%

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

      1. pow1 56.5%

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

   5. Applied egg-rr 56.5%

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

      1. unpow1 56.5%

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

      2. associate-*l* 47.7%

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

      3. fma-def 47.7%

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

      4. associate-*r* 56.4%

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

      5. fma-udef 56.4%

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

      6. *-commutative 56.4%

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

      7. associate-*r/ 56.4%

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

      8. unpow2 56.4%

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

      9. +-commutative 56.4%

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

      10. fma-def 56.4%

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

      11. +-commutative 56.4%

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

      12. unpow2 56.4%

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

      13. associate-*r/ 56.4%

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

      14. fma-def 56.4%

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

   7. Simplified 56.4%

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

      1. fma-udef 56.4%

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

   9. Applied egg-rr 56.4%

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

   10. Taylor expanded in U* around inf 38.8%

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

       1. associate-*r* 39.0%

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

       2. unpow2 39.0%

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

       3. times-frac 56.5%

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

       4. associate-*l/ 56.5%

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

       5. unpow2 56.5%

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

   12. Simplified 56.5%

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

   if 3.7999999999999997e-43 < l < 2.8000000000000001e-37

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in l around 0 99.2%

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

      1. associate-*r* 51.1%

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

      2. *-commutative 51.1%

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

   6. Simplified 51.1%

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

      1. *-un-lft-identity 51.1%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*l* 99.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   8. Applied egg-rr 99.2%

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

      1. *-lft-identity 99.2%

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

      2. associate-*r* 99.2%

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

   10. Simplified 99.2%

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

       1. sqrt-prod 99.2%

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

       2. *-commutative 99.2%

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

   12. Applied egg-rr 99.2%

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

   if 2.8000000000000001e-37 < l < 4.29999999999999988e-26

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in n around inf 53.3%

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

      1. associate-/l* 53.3%

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

      2. associate-*r/ 53.3%

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

      3. unpow2 53.3%

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

      4. associate-*r* 53.3%

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

      5. unpow2 53.3%

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

      6. unpow2 53.3%

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

   6. Simplified 53.3%

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

      1. *-un-lft-identity 53.3%

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

      2. associate-/r/ 53.3%

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

      3. *-commutative 53.3%

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

      4. associate-*l* 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-lft-identity 53.3%

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

      2. times-frac 53.3%

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

      3. associate-*l* 53.3%

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

   10. Simplified 53.3%

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

   if 4.29999999999999988e-26 < l < 8.80000000000000063e56 or 2.79999999999999987e151 < l < 2.6999999999999999e239

   1. Initial program 32.6%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in n around 0 34.9%

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

      1. metadata-eval 34.9%

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

      2. cancel-sign-sub-inv 34.9%

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

      3. cancel-sign-sub-inv 34.9%

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

      4. metadata-eval 34.9%

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

      5. +-commutative 34.9%

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

      6. unpow2 34.9%

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

      7. associate-*r/ 48.4%

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

      8. *-commutative 48.4%

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

      9. fma-udef 48.4%

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

   6. Simplified 48.4%

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

   if 8.80000000000000063e56 < l < 3.9500000000000001e139

   1. Initial program 41.7%

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

   3. Simplified 41.4%

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

      1. pow1 41.4%

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

   5. Applied egg-rr 41.4%

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

      1. unpow1 41.4%

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

      2. associate-*l* 42.0%

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

      3. fma-def 42.0%

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

      4. associate-*r* 48.5%

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

      5. fma-udef 48.5%

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

      6. *-commutative 48.5%

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

      7. associate-*r/ 48.4%

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

      8. unpow2 48.4%

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

      9. +-commutative 48.4%

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

      10. fma-def 48.4%

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

      11. +-commutative 48.4%

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

      12. unpow2 48.4%

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

      13. associate-*r/ 48.5%

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

      14. fma-def 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in l around 0 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. associate-*r/ 55.1%

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

      4. metadata-eval 55.1%

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

   10. Simplified 55.1%

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

   if 3.9500000000000001e139 < l < 2.79999999999999987e151 or 2.6999999999999999e239 < l

   1. Initial program 14.8%

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

   3. Simplified 21.2%

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

      1. sqrt-prod 21.2%

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

   5. Applied egg-rr 21.2%

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

   6. Taylor expanded in l around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-*r* 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/l* 79.9%

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

      5. unpow2 79.9%

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

      6. associate-*r/ 79.9%

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

      7. metadata-eval 79.9%

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

   8. Simplified 79.9%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{U \cdot U*}{Om} \cdot \left(n \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.95 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+151} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+239}\right):\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U* - U}} - \frac{2}{Om}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \end{array} \]

Alternative 5: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{if}\;\ell \leq 2.4 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{U \cdot U*}{Om} \cdot \left(n \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+151} \lor \neg \left(\ell \leq 5.5 \cdot 10^{+216}\right):\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))))
   (if (<= l 2.4e-80)
     (sqrt (* 2.0 (* t (* n U))))
     (if (<= l 3.5e-43)
       (sqrt (* 2.0 (* n (* (/ (* U U*) Om) (* n (/ (* l l) Om))))))
       (if (<= l 3.2e-38)
         (* (sqrt (* 2.0 n)) (sqrt (* U t)))
         (if (<= l 4.8e-26)
           (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (* l l) (* n (* n (- U U*))))))
           (if (<= l 8.8e+56)
             t_1
             (if (<= l 8.2e+85)
               (sqrt
                (*
                 2.0
                 (*
                  n
                  (*
                   U
                   (+
                    t
                    (*
                     (* l l)
                     (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))
               (if (or (<= l 1.3e+151) (not (<= l 5.5e+216)))
                 (*
                  (sqrt
                   (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)))))
                  (* l (sqrt 2.0)))
                 t_1)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	double tmp;
	if (l <= 2.4e-80) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else if (l <= 3.5e-43) {
		tmp = sqrt((2.0 * (n * (((U * U_42_) / Om) * (n * ((l * l) / Om))))));
	} else if (l <= 3.2e-38) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (l <= 4.8e-26) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((l * l) * (n * (n * (U - U_42_))))));
	} else if (l <= 8.8e+56) {
		tmp = t_1;
	} else if (l <= 8.2e+85) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	} else if ((l <= 1.3e+151) || !(l <= 5.5e+216)) {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))))
	tmp = 0.0
	if (l <= 2.4e-80)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	elseif (l <= 3.5e-43)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(Float64(U * U_42_) / Om) * Float64(n * Float64(Float64(l * l) / Om))))));
	elseif (l <= 3.2e-38)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (l <= 4.8e-26)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(l * l) * Float64(n * Float64(n * Float64(U - U_42_))))));
	elseif (l <= 8.8e+56)
		tmp = t_1;
	elseif (l <= 8.2e+85)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))));
	elseif ((l <= 1.3e+151) || !(l <= 5.5e+216))
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))) * Float64(l * sqrt(2.0)));
	else
		tmp = 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[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 2.4e-80], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.5e-43], N[Sqrt[N[(2.0 * N[(n * N[(N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.2e-38], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.8e-26], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(n * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.8e+56], t$95$1, If[LessEqual[l, 8.2e+85], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 1.3e+151], N[Not[LessEqual[l, 5.5e+216]], $MachinePrecision]], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < 2.3999999999999999e-80

   1. Initial program 49.7%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in l around 0 40.3%

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

      1. associate-*r* 41.4%

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

      2. *-commutative 41.4%

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

   6. Simplified 41.4%

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

   if 2.3999999999999999e-80 < l < 3.49999999999999997e-43

   1. Initial program 65.1%

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

   3. Simplified 56.5%

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

      1. pow1 56.5%

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

   5. Applied egg-rr 56.5%

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

      1. unpow1 56.5%

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

      2. associate-*l* 47.7%

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

      3. fma-def 47.7%

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

      4. associate-*r* 56.4%

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

      5. fma-udef 56.4%

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

      6. *-commutative 56.4%

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

      7. associate-*r/ 56.4%

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

      8. unpow2 56.4%

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

      9. +-commutative 56.4%

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

      10. fma-def 56.4%

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

      11. +-commutative 56.4%

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

      12. unpow2 56.4%

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

      13. associate-*r/ 56.4%

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

      14. fma-def 56.4%

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

   7. Simplified 56.4%

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

      1. fma-udef 56.4%

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

   9. Applied egg-rr 56.4%

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

   10. Taylor expanded in U* around inf 38.8%

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

       1. associate-*r* 39.0%

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

       2. unpow2 39.0%

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

       3. times-frac 56.5%

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

       4. associate-*l/ 56.5%

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

       5. unpow2 56.5%

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

   12. Simplified 56.5%

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

   if 3.49999999999999997e-43 < l < 3.19999999999999977e-38

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in l around 0 99.2%

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

      1. associate-*r* 51.1%

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

      2. *-commutative 51.1%

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

   6. Simplified 51.1%

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

      1. *-un-lft-identity 51.1%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*l* 99.2%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   8. Applied egg-rr 99.2%

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

      1. *-lft-identity 99.2%

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

      2. associate-*r* 99.2%

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

   10. Simplified 99.2%

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

       1. sqrt-prod 99.2%

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

       2. *-commutative 99.2%

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

   12. Applied egg-rr 99.2%

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

   if 3.19999999999999977e-38 < l < 4.8000000000000002e-26

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in n around inf 53.3%

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

      1. associate-/l* 53.3%

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

      2. associate-*r/ 53.3%

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

      3. unpow2 53.3%

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

      4. associate-*r* 53.3%

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

      5. unpow2 53.3%

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

      6. unpow2 53.3%

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

   6. Simplified 53.3%

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

      1. *-un-lft-identity 53.3%

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

      2. associate-/r/ 53.3%

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

      3. *-commutative 53.3%

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

      4. associate-*l* 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-lft-identity 53.3%

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

      2. times-frac 53.3%

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

      3. associate-*l* 53.3%

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

   10. Simplified 53.3%

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

   if 4.8000000000000002e-26 < l < 8.80000000000000063e56 or 1.30000000000000007e151 < l < 5.5e216

   1. Initial program 37.7%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in n around 0 40.1%

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

      1. metadata-eval 40.1%

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

      2. cancel-sign-sub-inv 40.1%

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

      3. cancel-sign-sub-inv 40.1%

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

      4. metadata-eval 40.1%

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

      5. +-commutative 40.1%

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

      6. unpow2 40.1%

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

      7. associate-*r/ 51.9%

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

      8. *-commutative 51.9%

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

      9. fma-udef 51.9%

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

   6. Simplified 51.9%

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

   if 8.80000000000000063e56 < l < 8.19999999999999957e85

   1. Initial program 68.2%

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

   3. Simplified 66.7%

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

      1. pow1 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. unpow1 66.7%

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

      2. associate-*l* 67.4%

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

      3. fma-def 67.4%

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

      4. associate-*r* 100.0%

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

      5. fma-udef 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r/ 99.5%

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

      8. unpow2 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-def 99.5%

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

      11. +-commutative 99.5%

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

      12. unpow2 99.5%

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

      13. associate-*r/ 100.0%

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

      14. fma-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

      2. unpow2 99.5%

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

      3. associate-*r/ 99.5%

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

      4. metadata-eval 99.5%

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

   10. Simplified 99.5%

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

   if 8.19999999999999957e85 < l < 1.30000000000000007e151 or 5.5e216 < l

   1. Initial program 20.9%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in l around inf 56.7%

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

      1. sub-neg 56.7%

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

      2. unpow2 56.7%

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

      3. times-frac 72.3%

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

      4. associate-*r/ 72.3%

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

      5. metadata-eval 72.3%

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

      6. distribute-neg-frac 72.3%

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

      7. metadata-eval 72.3%

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

   6. Simplified 72.3%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\frac{U \cdot U*}{Om} \cdot \left(n \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+151} \lor \neg \left(\ell \leq 5.5 \cdot 10^{+216}\right):\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \end{array} \]

Alternative 6: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\mathsf{fma}\left(-2, t_1, t\right) + \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+151} \lor \neg \left(\ell \leq 5.8 \cdot 10^{+216}\right):\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_1, -2, t\right)\right)\right)}\\ \end{array} \end{array} \]

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))))
   (if (<= l 5.4e+57)
     (sqrt
      (*
       (* 2.0 U)
       (* n (+ (fma -2.0 t_1 t) (* (/ n Om) (/ (* (* l l) U*) Om))))))
     (if (or (<= l 4e+151) (not (<= l 5.8e+216)))
       (*
        (sqrt (* U (* n (+ (* (/ n Om) (/ (- U* U) Om)) (/ -2.0 Om)))))
        (* l (sqrt 2.0)))
       (sqrt (* 2.0 (* U (* n (fma t_1 -2.0 t)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double tmp;
	if (l <= 5.4e+57) {
		tmp = sqrt(((2.0 * U) * (n * (fma(-2.0, t_1, t) + ((n / Om) * (((l * l) * U_42_) / Om))))));
	} else if ((l <= 4e+151) || !(l <= 5.8e+216)) {
		tmp = sqrt((U * (n * (((n / Om) * ((U_42_ - U) / Om)) + (-2.0 / Om))))) * (l * sqrt(2.0));
	} else {
		tmp = sqrt((2.0 * (U * (n * fma(t_1, -2.0, t)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (l <= 5.4e+57)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(fma(-2.0, t_1, t) + Float64(Float64(n / Om) * Float64(Float64(Float64(l * l) * U_42_) / Om))))));
	elseif ((l <= 4e+151) || !(l <= 5.8e+216))
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om)) + Float64(-2.0 / Om))))) * Float64(l * sqrt(2.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(t_1, -2.0, t)))));
	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]}, If[LessEqual[l, 5.4e+57], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] + N[(N[(n / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l, 4e+151], N[Not[LessEqual[l, 5.8e+216]], $MachinePrecision]], N[(N[Sqrt[N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.3999999999999997e57

   1. Initial program 50.5%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in U around 0 46.7%

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

      1. associate-*r* 46.7%

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

      2. +-commutative 46.7%

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

      3. mul-1-neg 46.7%

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

      4. unsub-neg 46.7%

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

      5. associate-+l- 46.7%

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

      6. cancel-sign-sub-inv 46.7%

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

      7. metadata-eval 46.7%

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

      8. +-commutative 46.7%

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

      9. unpow2 46.7%

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

      10. associate-*r/ 47.1%

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

      11. fma-def 47.1%

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

   6. Simplified 50.6%

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

   if 5.3999999999999997e57 < l < 4.00000000000000007e151 or 5.8000000000000002e216 < l

   1. Initial program 25.1%

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

   3. Simplified 30.6%

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

   4. Taylor expanded in l around inf 57.7%

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

      1. sub-neg 57.7%

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

      2. unpow2 57.7%

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

      3. times-frac 71.8%

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

      4. associate-*r/ 71.8%

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

      5. metadata-eval 71.8%

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

      6. distribute-neg-frac 71.8%

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

      7. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

   if 4.00000000000000007e151 < l < 5.8000000000000002e216

   1. Initial program 13.1%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in n around 0 13.7%

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

      1. metadata-eval 13.7%

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

      2. cancel-sign-sub-inv 13.7%

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

      3. cancel-sign-sub-inv 13.7%

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

      4. metadata-eval 13.7%

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

      5. +-commutative 13.7%

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

      6. unpow2 13.7%

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

      7. associate-*r/ 50.6%

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

      8. *-commutative 50.6%

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

      9. fma-udef 50.6%

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

   6. Simplified 50.6%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) + \frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+151} \lor \neg \left(\ell \leq 5.8 \cdot 10^{+216}\right):\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{U* - U}{Om} + \frac{-2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \end{array} \]

Alternative 7: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(U* - U\right)}\\ t_2 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;Om \leq -4.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(t_2, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -6.2 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\ell}{\frac{Om}{n \cdot \sqrt{2}}} \cdot \left(-t_1\right)\\ \mathbf{elif}\;Om \leq -6.2 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_1 \cdot \frac{\ell}{\frac{Om}{n}}\right)\\ \mathbf{elif}\;Om \leq -9.8 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t_2 \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-266}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* U (- U* U)))) (t_2 (* l (/ l Om))))
   (if (<= Om -4.6e+49)
     (sqrt (* 2.0 (* U (* n (fma t_2 -2.0 t)))))
     (if (<= Om -6.2e-174)
       (sqrt
        (*
         2.0
         (*
          n
          (*
           U
           (+ t (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))
       (if (<= Om -4e-216)
         (* (/ l (/ Om (* n (sqrt 2.0)))) (- t_1))
         (if (<= Om -6.2e-235)
           (* (sqrt 2.0) (* t_1 (/ l (/ Om n))))
           (if (<= Om -9.8e-253)
             (sqrt (* 2.0 (* n (* U (* t_2 (/ U* (/ Om n)))))))
             (if (<= Om -1e-266)
               (* t_1 (/ (* (sqrt 2.0) (* n l)) Om))
               (if (<= Om 2.8e-20)
                 (sqrt
                  (*
                   (* (/ -2.0 Om) (/ U Om))
                   (* (- U U*) (* (* n l) (* n l)))))
                 (sqrt
                  (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l))))))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((U * (U_42_ - U)));
	double t_2 = l * (l / Om);
	double tmp;
	if (Om <= -4.6e+49) {
		tmp = sqrt((2.0 * (U * (n * fma(t_2, -2.0, t)))));
	} else if (Om <= -6.2e-174) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	} else if (Om <= -4e-216) {
		tmp = (l / (Om / (n * sqrt(2.0)))) * -t_1;
	} else if (Om <= -6.2e-235) {
		tmp = sqrt(2.0) * (t_1 * (l / (Om / n)));
	} else if (Om <= -9.8e-253) {
		tmp = sqrt((2.0 * (n * (U * (t_2 * (U_42_ / (Om / n)))))));
	} else if (Om <= -1e-266) {
		tmp = t_1 * ((sqrt(2.0) * (n * l)) / Om);
	} else if (Om <= 2.8e-20) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(U_42_ - U)))
	t_2 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (Om <= -4.6e+49)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(t_2, -2.0, t)))));
	elseif (Om <= -6.2e-174)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))));
	elseif (Om <= -4e-216)
		tmp = Float64(Float64(l / Float64(Om / Float64(n * sqrt(2.0)))) * Float64(-t_1));
	elseif (Om <= -6.2e-235)
		tmp = Float64(sqrt(2.0) * Float64(t_1 * Float64(l / Float64(Om / n))));
	elseif (Om <= -9.8e-253)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t_2 * Float64(U_42_ / Float64(Om / n)))))));
	elseif (Om <= -1e-266)
		tmp = Float64(t_1 * Float64(Float64(sqrt(2.0) * Float64(n * l)) / Om));
	elseif (Om <= 2.8e-20)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(U - U_42_) * Float64(Float64(n * l) * Float64(n * l)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	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[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -4.6e+49], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t$95$2 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -6.2e-174], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -4e-216], N[(N[(l / N[(Om / N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-t$95$1)), $MachinePrecision], If[LessEqual[Om, -6.2e-235], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, -9.8e-253], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t$95$2 * N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1e-266], N[(t$95$1 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 2.8e-20], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if Om < -4.60000000000000004e49

   1. Initial program 42.1%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in n around 0 43.7%

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

      1. metadata-eval 43.7%

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

      2. cancel-sign-sub-inv 43.7%

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

      3. cancel-sign-sub-inv 43.7%

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

      4. metadata-eval 43.7%

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

      5. +-commutative 43.7%

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

      6. unpow2 43.7%

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

      7. associate-*r/ 54.3%

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

      8. *-commutative 54.3%

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

      9. fma-udef 54.3%

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

   6. Simplified 54.3%

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

   if -4.60000000000000004e49 < Om < -6.1999999999999998e-174

   1. Initial program 49.0%

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

   3. Simplified 44.6%

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

      1. pow1 44.6%

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

   5. Applied egg-rr 44.6%

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

      1. unpow1 44.6%

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

      2. associate-*l* 49.0%

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

      3. fma-def 49.0%

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

      4. associate-*r* 59.6%

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

      5. fma-udef 59.6%

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

      6. *-commutative 59.6%

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

      7. associate-*r/ 57.4%

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

      8. unpow2 57.4%

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

      9. +-commutative 57.4%

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

      10. fma-def 57.5%

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

      11. +-commutative 57.5%

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

      12. unpow2 57.5%

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

      13. associate-*r/ 59.6%

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

      14. fma-def 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in l around 0 66.3%

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

      1. unpow2 66.3%

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

      2. unpow2 66.3%

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

      3. associate-*r/ 66.3%

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

      4. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

   if -6.1999999999999998e-174 < Om < -4.0000000000000002e-216

   1. Initial program 36.1%

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

   3. Simplified 24.8%

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

      1. pow1 24.8%

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

   5. Applied egg-rr 24.8%

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

      1. unpow1 24.8%

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

      2. associate-*l* 25.6%

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

      3. fma-def 25.6%

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

      4. associate-*r* 47.2%

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

      5. fma-udef 47.2%

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

      6. *-commutative 47.2%

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

      7. associate-*r/ 47.2%

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

      8. unpow2 47.2%

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

      9. +-commutative 47.2%

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

      10. fma-def 47.2%

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

      11. +-commutative 47.2%

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

      12. unpow2 47.2%

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

      13. associate-*r/ 47.2%

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

      14. fma-def 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in n around inf 14.0%

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

      1. unpow2 14.0%

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

      2. times-frac 36.0%

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

      3. unpow2 36.0%

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

      4. associate-*r/ 46.5%

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

   10. Simplified 46.5%

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

   11. Taylor expanded in n around -inf 44.6%

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

       1. mul-1-neg 44.6%

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

       2. associate-/l* 55.1%

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

   13. Simplified 55.1%

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

   if -4.0000000000000002e-216 < Om < -6.2e-235

   1. Initial program 26.3%

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

   3. Simplified 26.3%

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

      1. sqrt-prod 26.3%

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

   5. Applied egg-rr 26.3%

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

   6. Taylor expanded in n around inf 45.5%

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

      1. associate-/l* 50.2%

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

   8. Simplified 50.2%

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

   if -6.2e-235 < Om < -9.7999999999999999e-253

   1. Initial program 27.2%

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

   3. Simplified 27.2%

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

      1. pow1 27.2%

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

   5. Applied egg-rr 27.2%

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

      1. unpow1 27.2%

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

      2. associate-*l* 50.8%

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

      3. fma-def 50.8%

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

      4. associate-*r* 51.2%

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

      5. fma-udef 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-*r/ 51.2%

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

      8. unpow2 51.2%

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

      9. +-commutative 51.2%

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

      10. fma-def 51.2%

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

      11. +-commutative 51.2%

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

      12. unpow2 51.2%

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

      13. associate-*r/ 51.2%

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

      14. fma-def 51.2%

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

   7. Simplified 51.2%

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

   8. Taylor expanded in n around inf 26.4%

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

      1. unpow2 26.4%

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

      2. times-frac 28.0%

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

      3. unpow2 28.0%

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

      4. associate-*r/ 51.6%

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

   10. Simplified 51.6%

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

   11. Taylor expanded in U* around inf 51.6%

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

       1. associate-/l* 51.6%

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

   13. Simplified 51.6%

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

   if -9.7999999999999999e-253 < Om < -9.9999999999999998e-267

   1. Initial program 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in n around inf 78.9%

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

      1. associate-*r* 79.7%

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

   6. Simplified 79.7%

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

   if -9.9999999999999998e-267 < Om < 2.8000000000000003e-20

   1. Initial program 44.5%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in n around inf 42.2%

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

      1. associate-/l* 40.3%

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

      2. associate-*r/ 40.3%

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

      3. unpow2 40.3%

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

      4. associate-*r* 40.3%

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

      5. unpow2 40.3%

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

      6. unpow2 40.3%

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

   6. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. associate-/r/ 42.1%

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

      3. *-commutative 42.1%

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

      4. associate-*l* 42.2%

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

   8. Applied egg-rr 42.2%

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

      1. *-lft-identity 42.2%

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

      2. times-frac 48.2%

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

      3. unpow2 48.2%

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

      4. unpow2 48.2%

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

      5. associate-*r* 48.1%

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

      6. unpow2 48.1%

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

      7. unpow2 48.1%

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

      8. swap-sqr 56.6%

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

   10. Simplified 56.6%

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

   if 2.8000000000000003e-20 < Om

   1. Initial program 54.5%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in Om around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-*r/ 52.4%

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

      3. associate-*r* 52.4%

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

   6. Simplified 52.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]
3. Recombined 8 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -6.2 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -4 \cdot 10^{-216}:\\ \;\;\;\;\frac{\ell}{\frac{Om}{n \cdot \sqrt{2}}} \cdot \left(-\sqrt{U \cdot \left(U* - U\right)}\right)\\ \mathbf{elif}\;Om \leq -6.2 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right)\\ \mathbf{elif}\;Om \leq -9.8 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{elif}\;Om \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 8: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -4 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.45 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right)\\ \mathbf{elif}\;Om \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -4e+46)
   (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
   (if (<= Om -1.45e-183)
     (sqrt
      (*
       2.0
       (*
        n
        (* U (+ t (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))
     (if (<= Om -1.1e-266)
       (* (sqrt 2.0) (* (sqrt (* U (- U* U))) (/ l (/ Om n))))
       (if (<= Om 1.25e-21)
         (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (- U U*) (* (* n l) (* n l)))))
         (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -4e+46) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else if (Om <= -1.45e-183) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	} else if (Om <= -1.1e-266) {
		tmp = sqrt(2.0) * (sqrt((U * (U_42_ - U))) * (l / (Om / n)));
	} else if (Om <= 1.25e-21) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -4e+46)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	elseif (Om <= -1.45e-183)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))));
	elseif (Om <= -1.1e-266)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(l / Float64(Om / n))));
	elseif (Om <= 1.25e-21)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(U - U_42_) * Float64(Float64(n * l) * Float64(n * l)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -4e+46], 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], If[LessEqual[Om, -1.45e-183], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.1e-266], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 1.25e-21], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if Om < -4e46

   1. Initial program 42.1%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in n around 0 43.7%

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

      1. metadata-eval 43.7%

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

      2. cancel-sign-sub-inv 43.7%

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

      3. cancel-sign-sub-inv 43.7%

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

      4. metadata-eval 43.7%

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

      5. +-commutative 43.7%

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

      6. unpow2 43.7%

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

      7. associate-*r/ 54.3%

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

      8. *-commutative 54.3%

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

      9. fma-udef 54.3%

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

   6. Simplified 54.3%

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

   if -4e46 < Om < -1.45e-183

   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 44.0%

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

      1. pow1 44.0%

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

   5. Applied egg-rr 44.0%

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

      1. unpow1 44.0%

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

      2. associate-*l* 48.2%

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

      3. fma-def 48.2%

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

      4. associate-*r* 60.1%

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

      5. fma-udef 60.1%

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

      6. *-commutative 60.1%

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

      7. associate-*r/ 58.1%

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

      8. unpow2 58.1%

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

      9. +-commutative 58.1%

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

      10. fma-def 58.1%

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

      11. +-commutative 58.1%

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

      12. unpow2 58.1%

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

      13. associate-*r/ 60.1%

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

      14. fma-def 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in l around 0 64.4%

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

      1. unpow2 64.4%

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

      2. unpow2 64.4%

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

      3. associate-*r/ 64.4%

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

      4. metadata-eval 64.4%

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

   10. Simplified 64.4%

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

   if -1.45e-183 < Om < -1.1e-266

   1. Initial program 23.7%

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

   3. Simplified 23.5%

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

      1. sqrt-prod 23.6%

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

   5. Applied egg-rr 23.6%

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

   6. Taylor expanded in n around inf 41.9%

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

      1. associate-/l* 43.0%

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

   8. Simplified 43.0%

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

   if -1.1e-266 < Om < 1.24999999999999993e-21

   1. Initial program 44.5%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in n around inf 42.2%

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

      1. associate-/l* 40.3%

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

      2. associate-*r/ 40.3%

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

      3. unpow2 40.3%

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

      4. associate-*r* 40.3%

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

      5. unpow2 40.3%

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

      6. unpow2 40.3%

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

   6. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. associate-/r/ 42.1%

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

      3. *-commutative 42.1%

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

      4. associate-*l* 42.2%

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

   8. Applied egg-rr 42.2%

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

      1. *-lft-identity 42.2%

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

      2. times-frac 48.2%

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

      3. unpow2 48.2%

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

      4. unpow2 48.2%

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

      5. associate-*r* 48.1%

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

      6. unpow2 48.1%

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

      7. unpow2 48.1%

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

      8. swap-sqr 56.6%

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

   10. Simplified 56.6%

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

   if 1.24999999999999993e-21 < Om

   1. Initial program 54.5%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in Om around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-*r/ 52.4%

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

      3. associate-*r* 52.4%

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

   6. Simplified 52.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.45 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right)\\ \mathbf{elif}\;Om \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 9: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -7.7 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -4.2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{elif}\;Om \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -7.7e+49)
   (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
   (if (<= Om -4.2e-180)
     (sqrt
      (*
       2.0
       (*
        n
        (* U (+ t (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))
     (if (<= Om -1.1e-266)
       (* (sqrt (* U (- U* U))) (/ (* (sqrt 2.0) (* n l)) Om))
       (if (<= Om 3.4e-21)
         (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (- U U*) (* (* n l) (* n l)))))
         (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -7.7e+49) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else if (Om <= -4.2e-180) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	} else if (Om <= -1.1e-266) {
		tmp = sqrt((U * (U_42_ - U))) * ((sqrt(2.0) * (n * l)) / Om);
	} else if (Om <= 3.4e-21) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -7.7e+49)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	elseif (Om <= -4.2e-180)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))));
	elseif (Om <= -1.1e-266)
		tmp = Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(Float64(sqrt(2.0) * Float64(n * l)) / Om));
	elseif (Om <= 3.4e-21)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(U - U_42_) * Float64(Float64(n * l) * Float64(n * l)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -7.7e+49], 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], If[LessEqual[Om, -4.2e-180], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.1e-266], N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 3.4e-21], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if Om < -7.7e49

   1. Initial program 42.1%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in n around 0 43.7%

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

      1. metadata-eval 43.7%

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

      2. cancel-sign-sub-inv 43.7%

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

      3. cancel-sign-sub-inv 43.7%

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

      4. metadata-eval 43.7%

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

      5. +-commutative 43.7%

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

      6. unpow2 43.7%

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

      7. associate-*r/ 54.3%

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

      8. *-commutative 54.3%

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

      9. fma-udef 54.3%

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

   6. Simplified 54.3%

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

   if -7.7e49 < Om < -4.1999999999999997e-180

   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 44.0%

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

      1. pow1 44.0%

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

   5. Applied egg-rr 44.0%

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

      1. unpow1 44.0%

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

      2. associate-*l* 48.2%

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

      3. fma-def 48.2%

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

      4. associate-*r* 60.1%

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

      5. fma-udef 60.1%

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

      6. *-commutative 60.1%

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

      7. associate-*r/ 58.1%

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

      8. unpow2 58.1%

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

      9. +-commutative 58.1%

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

      10. fma-def 58.1%

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

      11. +-commutative 58.1%

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

      12. unpow2 58.1%

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

      13. associate-*r/ 60.1%

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

      14. fma-def 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in l around 0 64.4%

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

      1. unpow2 64.4%

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

      2. unpow2 64.4%

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

      3. associate-*r/ 64.4%

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

      4. metadata-eval 64.4%

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

   10. Simplified 64.4%

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

   if -4.1999999999999997e-180 < Om < -1.1e-266

   1. Initial program 23.7%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in n around inf 41.8%

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

      1. associate-*r* 41.9%

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

   6. Simplified 41.9%

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

   if -1.1e-266 < Om < 3.4e-21

   1. Initial program 44.5%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in n around inf 42.2%

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

      1. associate-/l* 40.3%

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

      2. associate-*r/ 40.3%

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

      3. unpow2 40.3%

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

      4. associate-*r* 40.3%

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

      5. unpow2 40.3%

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

      6. unpow2 40.3%

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

   6. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. associate-/r/ 42.1%

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

      3. *-commutative 42.1%

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

      4. associate-*l* 42.2%

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

   8. Applied egg-rr 42.2%

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

      1. *-lft-identity 42.2%

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

      2. times-frac 48.2%

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

      3. unpow2 48.2%

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

      4. unpow2 48.2%

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

      5. associate-*r* 48.1%

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

      6. unpow2 48.1%

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

      7. unpow2 48.1%

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

      8. swap-sqr 56.6%

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

   10. Simplified 56.6%

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

   if 3.4e-21 < Om

   1. Initial program 54.5%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in Om around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-*r/ 52.4%

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

      3. associate-*r* 52.4%

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

   6. Simplified 52.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7.7 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -4.2 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \frac{\sqrt{2} \cdot \left(n \cdot \ell\right)}{Om}\\ \mathbf{elif}\;Om \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 10: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -9.2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -9.2e+45)
   (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
   (if (<= Om -1.4e-180)
     (sqrt
      (*
       2.0
       (*
        n
        (* U (+ t (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))
     (if (<= Om -1.1e-266)
       (/ (* (sqrt (* U (- U* U))) (* l (* n (sqrt 2.0)))) Om)
       (if (<= Om 4.2e-22)
         (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (- U U*) (* (* n l) (* n l)))))
         (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -9.2e+45) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else if (Om <= -1.4e-180) {
		tmp = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	} else if (Om <= -1.1e-266) {
		tmp = (sqrt((U * (U_42_ - U))) * (l * (n * sqrt(2.0)))) / Om;
	} else if (Om <= 4.2e-22) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -9.2e+45)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	elseif (Om <= -1.4e-180)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))));
	elseif (Om <= -1.1e-266)
		tmp = Float64(Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(l * Float64(n * sqrt(2.0)))) / Om);
	elseif (Om <= 4.2e-22)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(U - U_42_) * Float64(Float64(n * l) * Float64(n * l)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -9.2e+45], 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], If[LessEqual[Om, -1.4e-180], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.1e-266], N[(N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], If[LessEqual[Om, 4.2e-22], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if Om < -9.20000000000000049e45

   1. Initial program 42.1%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in n around 0 43.7%

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

      1. metadata-eval 43.7%

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

      2. cancel-sign-sub-inv 43.7%

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

      3. cancel-sign-sub-inv 43.7%

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

      4. metadata-eval 43.7%

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

      5. +-commutative 43.7%

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

      6. unpow2 43.7%

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

      7. associate-*r/ 54.3%

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

      8. *-commutative 54.3%

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

      9. fma-udef 54.3%

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

   6. Simplified 54.3%

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

   if -9.20000000000000049e45 < Om < -1.39999999999999999e-180

   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 44.0%

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

      1. pow1 44.0%

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

   5. Applied egg-rr 44.0%

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

      1. unpow1 44.0%

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

      2. associate-*l* 48.2%

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

      3. fma-def 48.2%

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

      4. associate-*r* 60.1%

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

      5. fma-udef 60.1%

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

      6. *-commutative 60.1%

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

      7. associate-*r/ 58.1%

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

      8. unpow2 58.1%

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

      9. +-commutative 58.1%

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

      10. fma-def 58.1%

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

      11. +-commutative 58.1%

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

      12. unpow2 58.1%

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

      13. associate-*r/ 60.1%

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

      14. fma-def 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in l around 0 64.4%

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

      1. unpow2 64.4%

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

      2. unpow2 64.4%

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

      3. associate-*r/ 64.4%

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

      4. metadata-eval 64.4%

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

   10. Simplified 64.4%

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

   if -1.39999999999999999e-180 < Om < -1.1e-266

   1. Initial program 23.7%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in n around inf 41.8%

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

      1. associate-*l/ 42.0%

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

   6. Simplified 42.0%

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

   if -1.1e-266 < Om < 4.20000000000000016e-22

   1. Initial program 44.5%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in n around inf 42.2%

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

      1. associate-/l* 40.3%

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

      2. associate-*r/ 40.3%

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

      3. unpow2 40.3%

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

      4. associate-*r* 40.3%

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

      5. unpow2 40.3%

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

      6. unpow2 40.3%

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

   6. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. associate-/r/ 42.1%

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

      3. *-commutative 42.1%

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

      4. associate-*l* 42.2%

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

   8. Applied egg-rr 42.2%

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

      1. *-lft-identity 42.2%

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

      2. times-frac 48.2%

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

      3. unpow2 48.2%

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

      4. unpow2 48.2%

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

      5. associate-*r* 48.1%

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

      6. unpow2 48.1%

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

      7. unpow2 48.1%

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

      8. swap-sqr 56.6%

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

   10. Simplified 56.6%

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

   if 4.20000000000000016e-22 < Om

   1. Initial program 54.5%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in Om around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-*r/ 52.4%

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

      3. associate-*r* 52.4%

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

   6. Simplified 52.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -1.1 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 11: 51.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;Om \leq -7.7 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -6.4 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 100:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           2.0
           (*
            n
            (*
             U
             (+
              t
              (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))))
   (if (<= Om -7.7e+49)
     (sqrt (* 2.0 (* U (* n (fma (* l (/ l Om)) -2.0 t)))))
     (if (<= Om -6.4e-131)
       t_1
       (if (<= Om 2.1e-55)
         (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (- U U*) (* (* n l) (* n l)))))
         (if (<= Om 100.0)
           t_1
           (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	double tmp;
	if (Om <= -7.7e+49) {
		tmp = sqrt((2.0 * (U * (n * fma((l * (l / Om)), -2.0, t)))));
	} else if (Om <= -6.4e-131) {
		tmp = t_1;
	} else if (Om <= 2.1e-55) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else if (Om <= 100.0) {
		tmp = t_1;
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))))
	tmp = 0.0
	if (Om <= -7.7e+49)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(Float64(l * Float64(l / Om)), -2.0, t)))));
	elseif (Om <= -6.4e-131)
		tmp = t_1;
	elseif (Om <= 2.1e-55)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(U - U_42_) * Float64(Float64(n * l) * Float64(n * l)))));
	elseif (Om <= 100.0)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	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[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -7.7e+49], 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], If[LessEqual[Om, -6.4e-131], t$95$1, If[LessEqual[Om, 2.1e-55], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 100.0], t$95$1, N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -7.7e49

   1. Initial program 42.1%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in n around 0 43.7%

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

      1. metadata-eval 43.7%

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

      2. cancel-sign-sub-inv 43.7%

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

      3. cancel-sign-sub-inv 43.7%

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

      4. metadata-eval 43.7%

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

      5. +-commutative 43.7%

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

      6. unpow2 43.7%

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

      7. associate-*r/ 54.3%

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

      8. *-commutative 54.3%

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

      9. fma-udef 54.3%

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

   6. Simplified 54.3%

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

   if -7.7e49 < Om < -6.3999999999999999e-131 or 2.1000000000000002e-55 < Om < 100

   1. Initial program 50.9%

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

   3. Simplified 45.2%

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

      1. pow1 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. unpow1 45.2%

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

      2. associate-*l* 51.2%

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

      3. fma-def 51.2%

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

      4. associate-*r* 61.9%

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

      5. fma-udef 61.9%

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

      6. *-commutative 61.9%

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

      7. associate-*r/ 60.1%

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

      8. unpow2 60.1%

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

      9. +-commutative 60.1%

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

      10. fma-def 60.2%

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

      11. +-commutative 60.2%

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

      12. unpow2 60.2%

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

      13. associate-*r/ 62.0%

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

      14. fma-def 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in l around 0 71.4%

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

      1. unpow2 71.4%

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

      2. unpow2 71.4%

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

      3. associate-*r/ 71.4%

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

      4. metadata-eval 71.4%

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

   10. Simplified 71.4%

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

   if -6.3999999999999999e-131 < Om < 2.1000000000000002e-55

   1. Initial program 39.2%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in n around inf 32.6%

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

      1. associate-/l* 31.3%

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

      2. associate-*r/ 31.3%

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

      3. unpow2 31.3%

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

      4. associate-*r* 31.3%

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

      5. unpow2 31.3%

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

      6. unpow2 31.3%

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

   6. Simplified 31.3%

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

      1. *-un-lft-identity 31.3%

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

      2. associate-/r/ 32.5%

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

      3. *-commutative 32.5%

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

      4. associate-*l* 32.6%

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

   8. Applied egg-rr 32.6%

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

      1. *-lft-identity 32.6%

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

      2. times-frac 39.7%

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

      3. unpow2 39.7%

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

      4. unpow2 39.7%

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

      5. associate-*r* 38.3%

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

      6. unpow2 38.3%

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

      7. unpow2 38.3%

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

      8. swap-sqr 44.6%

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

   10. Simplified 44.6%

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

   if 100 < Om

   1. Initial program 54.2%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in Om around inf 46.6%

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

      1. unpow2 46.6%

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

      2. associate-*r/ 53.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

      3. associate-*r* 53.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   6. Simplified 53.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7.7 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -6.4 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 100:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 12: 51.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{if}\;Om \leq -3.1 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -4.4 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.12 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 115:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l))))))
        (t_2
         (sqrt
          (*
           2.0
           (*
            n
            (*
             U
             (+
              t
              (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))))
   (if (<= Om -3.1e+48)
     t_1
     (if (<= Om -4.4e-130)
       t_2
       (if (<= Om 1.12e-54)
         (sqrt (* (* (/ -2.0 Om) (/ U Om)) (* (- U U*) (* (* n l) (* n l)))))
         (if (<= Om 115.0) t_2 t_1))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	double t_2 = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	double tmp;
	if (Om <= -3.1e+48) {
		tmp = t_1;
	} else if (Om <= -4.4e-130) {
		tmp = t_2;
	} else if (Om <= 1.12e-54) {
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else if (Om <= 115.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * (t - ((l / om) * (2.0d0 * l)))))
    t_2 = sqrt((2.0d0 * (n * (u * (t + ((l * l) * (((n * (u_42 - u)) / (om * om)) - (2.0d0 / om))))))))
    if (om <= (-3.1d+48)) then
        tmp = t_1
    else if (om <= (-4.4d-130)) then
        tmp = t_2
    else if (om <= 1.12d-54) then
        tmp = sqrt(((((-2.0d0) / om) * (u / om)) * ((u - u_42) * ((n * l) * (n * l)))))
    else if (om <= 115.0d0) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	double t_2 = Math.sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	double tmp;
	if (Om <= -3.1e+48) {
		tmp = t_1;
	} else if (Om <= -4.4e-130) {
		tmp = t_2;
	} else if (Om <= 1.12e-54) {
		tmp = Math.sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	} else if (Om <= 115.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))))
	t_2 = math.sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))))
	tmp = 0
	if Om <= -3.1e+48:
		tmp = t_1
	elif Om <= -4.4e-130:
		tmp = t_2
	elif Om <= 1.12e-54:
		tmp = math.sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))))
	elif Om <= 115.0:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))))
	t_2 = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om))))))))
	tmp = 0.0
	if (Om <= -3.1e+48)
		tmp = t_1;
	elseif (Om <= -4.4e-130)
		tmp = t_2;
	elseif (Om <= 1.12e-54)
		tmp = sqrt(Float64(Float64(Float64(-2.0 / Om) * Float64(U / Om)) * Float64(Float64(U - U_42_) * Float64(Float64(n * l) * Float64(n * l)))));
	elseif (Om <= 115.0)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	t_2 = sqrt((2.0 * (n * (U * (t + ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))))))));
	tmp = 0.0;
	if (Om <= -3.1e+48)
		tmp = t_1;
	elseif (Om <= -4.4e-130)
		tmp = t_2;
	elseif (Om <= 1.12e-54)
		tmp = sqrt((((-2.0 / Om) * (U / Om)) * ((U - U_42_) * ((n * l) * (n * l)))));
	elseif (Om <= 115.0)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -3.1e+48], t$95$1, If[LessEqual[Om, -4.4e-130], t$95$2, If[LessEqual[Om, 1.12e-54], N[Sqrt[N[(N[(N[(-2.0 / Om), $MachinePrecision] * N[(U / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 115.0], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -3.10000000000000005e48 or 115 < Om

   1. Initial program 47.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 49.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 44.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]
   5. Step-by-step derivation

      1. unpow2 44.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

      2. associate-*r/ 53.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

      3. associate-*r* 53.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   6. Simplified 53.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   if -3.10000000000000005e48 < Om < -4.3999999999999997e-130 or 1.11999999999999994e-54 < Om < 115

   1. Initial program 50.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 45.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 45.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 45.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 51.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. fma-def 51.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\right)\right)} \]

      4. associate-*r* 61.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)} + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      5. fma-udef 61.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t\right)}\right)\right)\right)} \]

      6. *-commutative 61.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)\right)} \]

      7. associate-*r/ 60.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right)\right)\right)\right)} \]

      8. unpow2 60.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om} + t\right)\right)\right)\right)} \]

      9. +-commutative 60.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)\right)} \]

      10. fma-def 60.2%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      11. +-commutative 60.2%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om} + t}\right)\right)\right)} \]

      12. unpow2 60.2%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right)\right)\right)} \]

      13. associate-*r/ 62.0%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)} \]

      14. fma-def 62.0%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\right)\right)\right)} \]

   7. Simplified 62.0%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in l around 0 71.4%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + {\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. unpow2 71.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      3. associate-*r/ 71.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      4. metadata-eval 71.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

   10. Simplified 71.4%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\right)\right)} \]

   if -4.3999999999999997e-130 < Om < 1.11999999999999994e-54

   1. Initial program 39.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 37.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in n around inf 32.6%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 31.3%

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{U}{\frac{{Om}^{2}}{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}}} \]

      2. associate-*r/ 31.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-2 \cdot U}{\frac{{Om}^{2}}{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}}} \]

      3. unpow2 31.3%

         \[\leadsto \sqrt{\frac{-2 \cdot U}{\frac{\color{blue}{Om \cdot Om}}{{\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}}} \]

      4. associate-*r* 31.3%

         \[\leadsto \sqrt{\frac{-2 \cdot U}{\frac{Om \cdot Om}{\color{blue}{\left({\ell}^{2} \cdot {n}^{2}\right) \cdot \left(U - U*\right)}}}} \]

      5. unpow2 31.3%

         \[\leadsto \sqrt{\frac{-2 \cdot U}{\frac{Om \cdot Om}{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {n}^{2}\right) \cdot \left(U - U*\right)}}} \]

      6. unpow2 31.3%

         \[\leadsto \sqrt{\frac{-2 \cdot U}{\frac{Om \cdot Om}{\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot n\right)}\right) \cdot \left(U - U*\right)}}} \]

   6. Simplified 31.3%

      \[\leadsto \sqrt{\color{blue}{\frac{-2 \cdot U}{\frac{Om \cdot Om}{\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 31.3%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{-2 \cdot U}{\frac{Om \cdot Om}{\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)}}}} \]

      2. associate-/r/ 32.5%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{-2 \cdot U}{Om \cdot Om} \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)\right)}} \]

      3. *-commutative 32.5%

         \[\leadsto 1 \cdot \sqrt{\frac{\color{blue}{U \cdot -2}}{Om \cdot Om} \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* 32.6%

         \[\leadsto 1 \cdot \sqrt{\frac{U \cdot -2}{Om \cdot Om} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)\right)}} \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{U \cdot -2}{Om \cdot Om} \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 32.6%

         \[\leadsto \color{blue}{\sqrt{\frac{U \cdot -2}{Om \cdot Om} \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. times-frac 39.7%

         \[\leadsto \sqrt{\color{blue}{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. unpow2 39.7%

         \[\leadsto \sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \left(\color{blue}{{\ell}^{2}} \cdot \left(\left(n \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      4. unpow2 39.7%

         \[\leadsto \sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \left({\ell}^{2} \cdot \left(\color{blue}{{n}^{2}} \cdot \left(U - U*\right)\right)\right)} \]

      5. associate-*r* 38.3%

         \[\leadsto \sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot {n}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. unpow2 38.3%

         \[\leadsto \sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {n}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. unpow2 38.3%

         \[\leadsto \sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot n\right)}\right) \cdot \left(U - U*\right)\right)} \]

      8. swap-sqr 44.6%

         \[\leadsto \sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \left(\color{blue}{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U - U*\right)\right)} \]

   10. Simplified 44.6%

       \[\leadsto \color{blue}{\sqrt{\left(\frac{U}{Om} \cdot \frac{-2}{Om}\right) \cdot \left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \left(U - U*\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -3.1 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{elif}\;Om \leq -4.4 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.12 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\left(\frac{-2}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 115:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 13: 48.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{if}\;Om \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -1.65 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -4.8 \cdot 10^{-172}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 2.95 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l)))))))
   (if (<= Om -2e+42)
     t_1
     (if (<= Om -1.65e-65)
       (sqrt
        (*
         2.0
         (* U (* (* l l) (* n (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om)))))))
       (if (<= Om -4.8e-172)
         (pow (* 2.0 (* U (* n t))) 0.5)
         (if (<= Om 2.95e-173)
           (sqrt (* 2.0 (* n (* U (* (* l (/ l Om)) (/ U* (/ Om 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 = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	double tmp;
	if (Om <= -2e+42) {
		tmp = t_1;
	} else if (Om <= -1.65e-65) {
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))));
	} else if (Om <= -4.8e-172) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (Om <= 2.95e-173) {
		tmp = sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * (t - ((l / om) * (2.0d0 * l)))))
    if (om <= (-2d+42)) then
        tmp = t_1
    else if (om <= (-1.65d-65)) then
        tmp = sqrt((2.0d0 * (u * ((l * l) * (n * (((n * (u_42 - u)) / (om * om)) - (2.0d0 / om)))))))
    else if (om <= (-4.8d-172)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (om <= 2.95d-173) then
        tmp = sqrt((2.0d0 * (n * (u * ((l * (l / om)) * (u_42 / (om / n)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	double tmp;
	if (Om <= -2e+42) {
		tmp = t_1;
	} else if (Om <= -1.65e-65) {
		tmp = Math.sqrt((2.0 * (U * ((l * l) * (n * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))));
	} else if (Om <= -4.8e-172) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (Om <= 2.95e-173) {
		tmp = Math.sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))))
	tmp = 0
	if Om <= -2e+42:
		tmp = t_1
	elif Om <= -1.65e-65:
		tmp = math.sqrt((2.0 * (U * ((l * l) * (n * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))))
	elif Om <= -4.8e-172:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif Om <= 2.95e-173:
		tmp = math.sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))))
	tmp = 0.0
	if (Om <= -2e+42)
		tmp = t_1;
	elseif (Om <= -1.65e-65)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(l * l) * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om)))))));
	elseif (Om <= -4.8e-172)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (Om <= 2.95e-173)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(Float64(l * Float64(l / Om)) * Float64(U_42_ / Float64(Om / n)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	tmp = 0.0;
	if (Om <= -2e+42)
		tmp = t_1;
	elseif (Om <= -1.65e-65)
		tmp = sqrt((2.0 * (U * ((l * l) * (n * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))));
	elseif (Om <= -4.8e-172)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (Om <= 2.95e-173)
		tmp = sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -2e+42], t$95$1, If[LessEqual[Om, -1.65e-65], N[Sqrt[N[(2.0 * N[(U * N[(N[(l * l), $MachinePrecision] * N[(n * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -4.8e-172], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[Om, 2.95e-173], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -2.00000000000000009e42 or 2.94999999999999998e-173 < Om

   1. Initial program 49.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 44.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]
   5. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

      2. associate-*r/ 51.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

      3. associate-*r* 51.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   6. Simplified 51.9%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   if -2.00000000000000009e42 < Om < -1.6500000000000001e-65

   1. Initial program 26.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 25.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

   4. Taylor expanded in l around inf 44.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 44.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. unpow2 44.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      3. associate-*r/ 44.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      4. metadata-eval 44.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

   6. Simplified 44.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}} \]

   if -1.6500000000000001e-65 < Om < -4.8000000000000002e-172

   1. Initial program 66.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 45.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 45.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   6. Simplified 45.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 58.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]

      2. associate-*l* 58.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]

   8. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   if -4.8000000000000002e-172 < Om < 2.94999999999999998e-173

   1. Initial program 31.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 27.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 27.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 27.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 29.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. fma-def 29.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\right)\right)} \]

      4. associate-*r* 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)} + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      5. fma-udef 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t\right)}\right)\right)\right)} \]

      6. *-commutative 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)\right)} \]

      7. associate-*r/ 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right)\right)\right)\right)} \]

      8. unpow2 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om} + t\right)\right)\right)\right)} \]

      9. +-commutative 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)\right)} \]

      10. fma-def 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      11. +-commutative 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om} + t}\right)\right)\right)} \]

      12. unpow2 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right)\right)\right)} \]

      13. associate-*r/ 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)} \]

      14. fma-def 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\right)\right)\right)} \]

   7. Simplified 34.3%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in n around inf 36.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)}{{Om}^{2}}}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 36.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      2. times-frac 40.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)} \]

      3. unpow2 40.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)} \]

      4. associate-*r/ 45.6%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)} \]

   10. Simplified 45.6%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)} \]

   11. Taylor expanded in U* around inf 45.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U* \cdot n}{Om}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-/l* 45.8%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U*}{\frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 45.8%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U*}{\frac{Om}{n}}}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{elif}\;Om \leq -1.65 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -4.8 \cdot 10^{-172}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 2.95 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 14: 48.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{if}\;Om \leq -2.5 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Om \leq -3.7 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -7 \cdot 10^{-173}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 8 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l)))))))
   (if (<= Om -2.5e+42)
     t_1
     (if (<= Om -3.7e-69)
       (sqrt
        (*
         2.0
         (* n (* U (* (* l l) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om)))))))
       (if (<= Om -7e-173)
         (pow (* 2.0 (* U (* n t))) 0.5)
         (if (<= Om 8e-173)
           (sqrt (* 2.0 (* n (* U (* (* l (/ l Om)) (/ U* (/ Om 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 = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	double tmp;
	if (Om <= -2.5e+42) {
		tmp = t_1;
	} else if (Om <= -3.7e-69) {
		tmp = sqrt((2.0 * (n * (U * ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))));
	} else if (Om <= -7e-173) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (Om <= 8e-173) {
		tmp = sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * (t - ((l / om) * (2.0d0 * l)))))
    if (om <= (-2.5d+42)) then
        tmp = t_1
    else if (om <= (-3.7d-69)) then
        tmp = sqrt((2.0d0 * (n * (u * ((l * l) * (((n * (u_42 - u)) / (om * om)) - (2.0d0 / om)))))))
    else if (om <= (-7d-173)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (om <= 8d-173) then
        tmp = sqrt((2.0d0 * (n * (u * ((l * (l / om)) * (u_42 / (om / n)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	double tmp;
	if (Om <= -2.5e+42) {
		tmp = t_1;
	} else if (Om <= -3.7e-69) {
		tmp = Math.sqrt((2.0 * (n * (U * ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))));
	} else if (Om <= -7e-173) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (Om <= 8e-173) {
		tmp = Math.sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))))
	tmp = 0
	if Om <= -2.5e+42:
		tmp = t_1
	elif Om <= -3.7e-69:
		tmp = math.sqrt((2.0 * (n * (U * ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))))
	elif Om <= -7e-173:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif Om <= 8e-173:
		tmp = math.sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))))
	tmp = 0.0
	if (Om <= -2.5e+42)
		tmp = t_1;
	elseif (Om <= -3.7e-69)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(Float64(l * l) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om)))))));
	elseif (Om <= -7e-173)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (Om <= 8e-173)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(Float64(l * Float64(l / Om)) * Float64(U_42_ / Float64(Om / n)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	tmp = 0.0;
	if (Om <= -2.5e+42)
		tmp = t_1;
	elseif (Om <= -3.7e-69)
		tmp = sqrt((2.0 * (n * (U * ((l * l) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))))));
	elseif (Om <= -7e-173)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (Om <= 8e-173)
		tmp = sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -2.5e+42], t$95$1, If[LessEqual[Om, -3.7e-69], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(l * l), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -7e-173], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[Om, 8e-173], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -2.50000000000000003e42 or 8.0000000000000003e-173 < Om

   1. Initial program 49.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 44.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]
   5. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

      2. associate-*r/ 51.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

      3. associate-*r* 51.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   6. Simplified 51.9%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   if -2.50000000000000003e42 < Om < -3.7000000000000002e-69

   1. Initial program 26.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 25.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 25.6%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 25.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 25.6%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 35.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. fma-def 35.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\right)\right)} \]

      4. associate-*r* 49.0%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)} + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      5. fma-udef 49.0%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t\right)}\right)\right)\right)} \]

      6. *-commutative 49.0%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)\right)} \]

      7. associate-*r/ 44.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right)\right)\right)\right)} \]

      8. unpow2 44.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om} + t\right)\right)\right)\right)} \]

      9. +-commutative 44.4%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)\right)} \]

      10. fma-def 44.5%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      11. +-commutative 44.5%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om} + t}\right)\right)\right)} \]

      12. unpow2 44.5%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right)\right)\right)} \]

      13. associate-*r/ 49.1%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)} \]

      14. fma-def 49.1%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\right)\right)\right)} \]

   7. Simplified 49.1%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in l around inf 44.8%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 44.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      2. unpow2 44.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)} \]

      3. associate-*r/ 44.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)\right)\right)} \]

      4. metadata-eval 44.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)\right)\right)\right)} \]

   10. Simplified 44.8%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)}\right)\right)} \]

   if -3.7000000000000002e-69 < Om < -7.00000000000000029e-173

   1. Initial program 66.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 45.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 45.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   6. Simplified 45.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 58.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]

      2. associate-*l* 58.3%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]

   8. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   if -7.00000000000000029e-173 < Om < 8.0000000000000003e-173

   1. Initial program 31.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 27.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 27.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 27.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 29.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. fma-def 29.8%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\right)\right)} \]

      4. associate-*r* 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)} + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      5. fma-udef 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t\right)}\right)\right)\right)} \]

      6. *-commutative 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)\right)} \]

      7. associate-*r/ 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right)\right)\right)\right)} \]

      8. unpow2 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om} + t\right)\right)\right)\right)} \]

      9. +-commutative 34.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)\right)} \]

      10. fma-def 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      11. +-commutative 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om} + t}\right)\right)\right)} \]

      12. unpow2 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right)\right)\right)} \]

      13. associate-*r/ 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)} \]

      14. fma-def 34.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\right)\right)\right)} \]

   7. Simplified 34.3%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in n around inf 36.1%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)}{{Om}^{2}}}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 36.1%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      2. times-frac 40.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)} \]

      3. unpow2 40.9%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)} \]

      4. associate-*r/ 45.6%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)} \]

   10. Simplified 45.6%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)} \]

   11. Taylor expanded in U* around inf 45.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U* \cdot n}{Om}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-/l* 45.8%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U*}{\frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 45.8%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U*}{\frac{Om}{n}}}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{elif}\;Om \leq -3.7 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq -7 \cdot 10^{-173}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 8 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 15: 49.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.3 \cdot 10^{-123} \lor \neg \left(Om \leq 6.6 \cdot 10^{-173}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.3e-123) (not (<= Om 6.6e-173)))
   (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l)))))
   (sqrt (* 2.0 (* n (* U (* (* l (/ l Om)) (/ 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 ((Om <= -1.3e-123) || !(Om <= 6.6e-173)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	} else {
		tmp = sqrt((2.0 * (n * (U * ((l * (l / Om)) * (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 ((om <= (-1.3d-123)) .or. (.not. (om <= 6.6d-173))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l / om) * (2.0d0 * l)))))
    else
        tmp = sqrt((2.0d0 * (n * (u * ((l * (l / om)) * (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 ((Om <= -1.3e-123) || !(Om <= 6.6e-173)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.3e-123) or not (Om <= 6.6e-173):
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * ((l * (l / Om)) * (U_42_ / (Om / n)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.3e-123) || !(Om <= 6.6e-173))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(Float64(l * Float64(l / Om)) * 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 ((Om <= -1.3e-123) || ~((Om <= 6.6e-173)))
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	else
		tmp = sqrt((2.0 * (n * (U * ((l * (l / Om)) * (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[Or[LessEqual[Om, -1.3e-123], N[Not[LessEqual[Om, 6.6e-173]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.29999999999999998e-123 or 6.6000000000000006e-173 < Om

   1. Initial program 48.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 43.7%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]
   5. Step-by-step derivation

      1. unpow2 43.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

      2. associate-*r/ 49.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

      3. associate-*r* 49.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   6. Simplified 49.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   if -1.29999999999999998e-123 < Om < 6.6000000000000006e-173

   1. Initial program 35.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 31.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. pow1 31.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]

   5. Applied egg-rr 31.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}^{1}}} \]
   6. Step-by-step derivation

      1. unpow1 31.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

      2. associate-*l* 35.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)}} \]

      3. fma-def 35.5%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\right)\right)} \]

      4. associate-*r* 39.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)} + \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)\right)} \]

      5. fma-udef 39.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t\right)}\right)\right)\right)} \]

      6. *-commutative 39.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)\right)} \]

      7. associate-*r/ 39.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right)\right)\right)\right)} \]

      8. unpow2 39.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(-2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om} + t\right)\right)\right)\right)} \]

      9. +-commutative 39.3%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)\right)} \]

      10. fma-def 39.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      11. +-commutative 39.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om} + t}\right)\right)\right)} \]

      12. unpow2 39.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right)\right)\right)} \]

      13. associate-*r/ 39.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right)\right)\right)} \]

      14. fma-def 39.3%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\right)\right)\right)} \]

   7. Simplified 39.3%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, U* - U, \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)\right)}} \]

   8. Taylor expanded in n around inf 37.2%

      \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)}{{Om}^{2}}}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 37.2%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      2. times-frac 41.5%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\frac{{\ell}^{2}}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)} \]

      3. unpow2 41.5%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)} \]

      4. associate-*r/ 45.5%

         \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)} \]

   10. Simplified 45.5%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \color{blue}{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)} \]

   11. Taylor expanded in U* around inf 45.7%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U* \cdot n}{Om}}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-/l* 45.8%

          \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U*}{\frac{Om}{n}}}\right)\right)\right)} \]

   13. Simplified 45.8%

       \[\leadsto \sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{U*}{\frac{Om}{n}}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.3 \cdot 10^{-123} \lor \neg \left(Om \leq 6.6 \cdot 10^{-173}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot \frac{U*}{\frac{Om}{n}}\right)\right)\right)}\\ \end{array} \]

Alternative 16: 46.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -2.5e+143)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* (* 2.0 (* n U)) (- t (* (/ l Om) (* 2.0 l)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.5e+143) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= (-2.5d+143)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((l / om) * (2.0d0 * l)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -2.5e+143) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -2.5e+143:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -2.5e+143)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(l / Om) * Float64(2.0 * l)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -2.5e+143)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * (n * U)) * (t - ((l / Om) * (2.0 * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -2.5e+143], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.50000000000000006e143

   1. Initial program 34.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 31.0%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 25.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 25.6%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   6. Simplified 25.6%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 31.4%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]

      2. associate-*l* 31.4%

         \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

   if -2.50000000000000006e143 < U*

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 44.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]
   5. Step-by-step derivation

      1. unpow2 44.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)} \]

      2. associate-*r/ 49.0%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)} \]

      3. associate-*r* 49.0%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]

   6. Simplified 49.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 17: 38.0% 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 45.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 34.4%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 34.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

6. Simplified 34.4%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
7. Step-by-step derivation

   1. pow1/2 35.9%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]

   2. associate-*l* 35.9%

      \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]

8. Applied egg-rr 35.9%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

9. Final simplification 35.9%

   \[\leadsto {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \]

Alternative 18: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* t (* n U)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (t * (n * U))));
}

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 * (t * (n * u))))
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 * (t * (n * U))));
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (t * (n * U))))

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(t * Float64(n * U))))
end

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (t * (n * U))));
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[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 45.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right)}} \]

4. Taylor expanded in l around 0 34.4%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 33.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   2. *-commutative 33.7%

      \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

6. Simplified 33.7%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

7. Final simplification 33.7%

   \[\leadsto \sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)} \]

Alternative 19: 36.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* 2.0 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(Float64(2.0 * 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[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 45.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 46.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 34.4%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 34.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

6. Simplified 34.4%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

7. Final simplification 34.4%

   \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \]

Reproduce

herbie shell --seed 2023278 
(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*))))))