Toniolo and Linder, Equation (13)

Percentage Accurate: 48.3% → 61.8%

Time: 25.7s

Alternatives: 20

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 48.3% 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: 61.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)\\ t_3 := {\left(\sqrt{2}\right)}^{0.3333333333333333}\\ \mathbf{if}\;t_2 \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(t_3 \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t + \left(\frac{n \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}^{3}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(t_3 \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

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
         (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))
        (t_3 (pow (sqrt 2.0) 0.3333333333333333)))
   (if (<= t_2 5e-323)
     (pow
      (*
       t_3
       (exp
        (*
         0.16666666666666666
         (+
          (log
           (*
            n
            (+
             t
             (-
              (/ (* n (* U* (pow l 2.0))) (pow Om 2.0))
              (* 2.0 (/ (pow l 2.0) Om))))))
          (log U)))))
      3.0)
     (if (<= t_2 INFINITY)
       (sqrt
        (pow
         (cbrt
          (* (* 2.0 (* n U)) (- t (fma 2.0 (* l (/ l Om)) (* t_1 (- U U*))))))
         3.0))
       (pow
        (*
         t_3
         (exp
          (*
           0.16666666666666666
           (+
            (log
             (*
              (* n U)
              (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 Om)))))
            (* -2.0 (log (/ 1.0 l)))))))
        3.0)))))

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 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double t_3 = pow(sqrt(2.0), 0.3333333333333333);
	double tmp;
	if (t_2 <= 5e-323) {
		tmp = pow((t_3 * exp((0.16666666666666666 * (log((n * (t + (((n * (U_42_ * pow(l, 2.0))) / pow(Om, 2.0)) - (2.0 * (pow(l, 2.0) / Om)))))) + log(U))))), 3.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(pow(cbrt(((2.0 * (n * U)) * (t - fma(2.0, (l * (l / Om)), (t_1 * (U - U_42_)))))), 3.0));
	} else {
		tmp = pow((t_3 * exp((0.16666666666666666 * (log(((n * U) * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))) + (-2.0 * log((1.0 / l))))))), 3.0);
	}
	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 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	t_3 = sqrt(2.0) ^ 0.3333333333333333
	tmp = 0.0
	if (t_2 <= 5e-323)
		tmp = Float64(t_3 * exp(Float64(0.16666666666666666 * Float64(log(Float64(n * Float64(t + Float64(Float64(Float64(n * Float64(U_42_ * (l ^ 2.0))) / (Om ^ 2.0)) - Float64(2.0 * Float64((l ^ 2.0) / Om)))))) + log(U))))) ^ 3.0;
	elseif (t_2 <= Inf)
		tmp = sqrt((cbrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(U - U_42_)))))) ^ 3.0));
	else
		tmp = Float64(t_3 * exp(Float64(0.16666666666666666 * Float64(log(Float64(Float64(n * U) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om))))) + Float64(-2.0 * log(Float64(1.0 / l))))))) ^ 3.0;
	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[(N[(N[(2.0 * n), $MachinePrecision] * U), $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]}, Block[{t$95$3 = N[Power[N[Sqrt[2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, If[LessEqual[t$95$2, 5e-323], N[Power[N[(t$95$3 * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(n * N[(t + N[(N[(N[(n * N[(U$42$ * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[Power[N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$3 * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.94066e-323

   1. Initial program 23.3%

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

   3. Simplified 23.4%

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

   5. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in U around 0 44.0%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}}^{3} \]

   if 4.94066e-323 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 68.7%

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

   3. Simplified 71.6%

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

      1. associate-*r* 71.6%

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

      2. fma-udef 71.6%

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

      3. associate-*l/ 64.3%

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

      4. associate-*r* 68.7%

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

      5. *-commutative 68.7%

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

      6. associate--l- 68.7%

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

      7. add-cube-cbrt 68.3%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}} \]

      8. pow3 68.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\right)}^{3}}} \]

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   5. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in l around inf 26.6%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left({\left(\sqrt{2}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t + \left(\frac{n \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}^{3}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{2}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 2: 61.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)\\ t_3 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t_2 \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log U + \log \left(n \cdot \left(t + \left(\frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}} - 2 \cdot t_3\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, t_3, t_1 \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{2}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

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
         (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))
        (t_3 (* l (/ l Om))))
   (if (<= t_2 5e-323)
     (pow
      (*
       (cbrt (sqrt 2.0))
       (pow
        (exp 0.16666666666666666)
        (+
         (log U)
         (log
          (* n (+ t (- (/ n (/ (* Om Om) (* (* l l) U*))) (* 2.0 t_3))))))))
      3.0)
     (if (<= t_2 INFINITY)
       (sqrt
        (pow
         (cbrt (* (* 2.0 (* n U)) (- t (fma 2.0 t_3 (* t_1 (- U U*))))))
         3.0))
       (pow
        (*
         (pow (sqrt 2.0) 0.3333333333333333)
         (exp
          (*
           0.16666666666666666
           (+
            (log
             (*
              (* n U)
              (+ (/ (* n (- U* U)) (pow Om 2.0)) (* 2.0 (/ -1.0 Om)))))
            (* -2.0 (log (/ 1.0 l)))))))
        3.0)))))

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 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double t_3 = l * (l / Om);
	double tmp;
	if (t_2 <= 5e-323) {
		tmp = pow((cbrt(sqrt(2.0)) * pow(exp(0.16666666666666666), (log(U) + log((n * (t + ((n / ((Om * Om) / ((l * l) * U_42_))) - (2.0 * t_3)))))))), 3.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(pow(cbrt(((2.0 * (n * U)) * (t - fma(2.0, t_3, (t_1 * (U - U_42_)))))), 3.0));
	} else {
		tmp = pow((pow(sqrt(2.0), 0.3333333333333333) * exp((0.16666666666666666 * (log(((n * U) * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om))))) + (-2.0 * log((1.0 / l))))))), 3.0);
	}
	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 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	t_3 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (t_2 <= 5e-323)
		tmp = Float64(cbrt(sqrt(2.0)) * (exp(0.16666666666666666) ^ Float64(log(U) + log(Float64(n * Float64(t + Float64(Float64(n / Float64(Float64(Om * Om) / Float64(Float64(l * l) * U_42_))) - Float64(2.0 * t_3)))))))) ^ 3.0;
	elseif (t_2 <= Inf)
		tmp = sqrt((cbrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - fma(2.0, t_3, Float64(t_1 * Float64(U - U_42_)))))) ^ 3.0));
	else
		tmp = Float64((sqrt(2.0) ^ 0.3333333333333333) * exp(Float64(0.16666666666666666 * Float64(log(Float64(Float64(n * U) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om))))) + Float64(-2.0 * log(Float64(1.0 / l))))))) ^ 3.0;
	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[(N[(N[(2.0 * n), $MachinePrecision] * U), $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]}, Block[{t$95$3 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-323], N[Power[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(N[Log[U], $MachinePrecision] + N[Log[N[(n * N[(t + N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[Power[N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * t$95$3 + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], N[Power[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.94066e-323

   1. Initial program 23.3%

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

   3. Simplified 23.4%

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

   5. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in U around 0 44.0%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 44.0%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{\sqrt{2} \cdot 1}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}^{3} \]

      2. *-rgt-identity 44.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{2}}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}^{3} \]

      3. exp-prod 42.0%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}}\right)}^{3} \]

   8. Simplified 42.0%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(n \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - \frac{n}{\frac{Om \cdot Om}{U* \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right) + \log U\right)}\right)}}^{3} \]

   if 4.94066e-323 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 68.7%

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

   3. Simplified 71.6%

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

      1. associate-*r* 71.6%

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

      2. fma-udef 71.6%

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

      3. associate-*l/ 64.3%

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

      4. associate-*r* 68.7%

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

      5. *-commutative 68.7%

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

      6. associate--l- 68.7%

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

      7. add-cube-cbrt 68.3%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}} \]

      8. pow3 68.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\right)}^{3}}} \]

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   5. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in l around inf 26.6%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log U + \log \left(n \cdot \left(t + \left(\frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}} - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{2}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 3: 62.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(U - U*\right)\right)\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t_2}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(U \cdot \left(-n\right)\right)\right) - -2 \cdot \log \ell\right)}\right)}^{3}\\ \end{array} \end{array} \]

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 (- t (fma 2.0 (* l (/ l Om)) (* t_1 (- U U*)))))
        (t_3
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U))))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t_2)))
     (if (<= t_3 INFINITY)
       (sqrt (pow (cbrt (* (* 2.0 (* n U)) t_2)) 3.0))
       (pow
        (*
         (cbrt (sqrt 2.0))
         (pow
          (exp 0.16666666666666666)
          (-
           (log (* (+ (/ n (/ (* Om Om) (- U U*))) (/ 2.0 Om)) (* U (- n))))
           (* -2.0 (log l)))))
        3.0)))))

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 = t - fma(2.0, (l * (l / Om)), (t_1 * (U - U_42_)));
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t_2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(pow(cbrt(((2.0 * (n * U)) * t_2)), 3.0));
	} else {
		tmp = pow((cbrt(sqrt(2.0)) * pow(exp(0.16666666666666666), (log((((n / ((Om * Om) / (U - U_42_))) + (2.0 / Om)) * (U * -n))) - (-2.0 * log(l))))), 3.0);
	}
	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 = Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(U - U_42_))))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t_2)));
	elseif (t_3 <= Inf)
		tmp = sqrt((cbrt(Float64(Float64(2.0 * Float64(n * U)) * t_2)) ^ 3.0));
	else
		tmp = Float64(cbrt(sqrt(2.0)) * (exp(0.16666666666666666) ^ Float64(log(Float64(Float64(Float64(n / Float64(Float64(Om * Om) / Float64(U - U_42_))) + Float64(2.0 / Om)) * Float64(U * Float64(-n)))) - Float64(-2.0 * log(l))))) ^ 3.0;
	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[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[Power[N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], N[Power[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(N[Log[N[(N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * (-n)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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 23.3%

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

      1. pow1/2 23.3%

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

      2. fma-udef 23.3%

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

      3. associate-*l/ 23.3%

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

      4. associate-*r* 23.3%

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

      5. *-commutative 23.3%

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

      6. associate--l- 23.3%

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

      7. associate-*r* 23.3%

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

      8. associate-*l* 38.9%

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

   5. Applied egg-rr 39.8%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/2 39.8%

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

   7. Simplified 39.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 68.5%

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

   3. Simplified 71.4%

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

      1. associate-*r* 71.4%

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

      2. fma-udef 71.4%

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

      3. associate-*l/ 64.1%

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

      4. associate-*r* 68.5%

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

      5. *-commutative 68.5%

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

      6. associate--l- 68.5%

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

      7. add-cube-cbrt 68.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}} \]

      8. pow3 68.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 75.3%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\right)}^{3}}} \]

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   5. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in l around inf 26.6%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 26.6%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{\sqrt{2} \cdot 1}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3} \]

      2. *-rgt-identity 26.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{2}}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3} \]

      3. exp-prod 26.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}}\right)}^{3} \]

   8. Simplified 26.1%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right)\right) + -2 \cdot \left(-\log \ell\right)\right)}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(U \cdot \left(-n\right)\right)\right) - -2 \cdot \log \ell\right)}\right)}^{3}\\ \end{array} \]

Alternative 4: 61.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)\\ t_3 := \sqrt[3]{\sqrt{2}}\\ t_4 := \ell \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t_2 \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(t_3 \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log U + \log \left(n \cdot \left(t + \left(\frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}} - 2 \cdot t_4\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, t_4, t_1 \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(t_3 \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(U \cdot \left(-n\right)\right)\right) - -2 \cdot \log \ell\right)}\right)}^{3}\\ \end{array} \end{array} \]

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
         (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))
        (t_3 (cbrt (sqrt 2.0)))
        (t_4 (* l (/ l Om))))
   (if (<= t_2 5e-323)
     (pow
      (*
       t_3
       (pow
        (exp 0.16666666666666666)
        (+
         (log U)
         (log
          (* n (+ t (- (/ n (/ (* Om Om) (* (* l l) U*))) (* 2.0 t_4))))))))
      3.0)
     (if (<= t_2 INFINITY)
       (sqrt
        (pow
         (cbrt (* (* 2.0 (* n U)) (- t (fma 2.0 t_4 (* t_1 (- U U*))))))
         3.0))
       (pow
        (*
         t_3
         (pow
          (exp 0.16666666666666666)
          (-
           (log (* (+ (/ n (/ (* Om Om) (- U U*))) (/ 2.0 Om)) (* U (- n))))
           (* -2.0 (log l)))))
        3.0)))))

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 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double t_3 = cbrt(sqrt(2.0));
	double t_4 = l * (l / Om);
	double tmp;
	if (t_2 <= 5e-323) {
		tmp = pow((t_3 * pow(exp(0.16666666666666666), (log(U) + log((n * (t + ((n / ((Om * Om) / ((l * l) * U_42_))) - (2.0 * t_4)))))))), 3.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(pow(cbrt(((2.0 * (n * U)) * (t - fma(2.0, t_4, (t_1 * (U - U_42_)))))), 3.0));
	} else {
		tmp = pow((t_3 * pow(exp(0.16666666666666666), (log((((n / ((Om * Om) / (U - U_42_))) + (2.0 / Om)) * (U * -n))) - (-2.0 * log(l))))), 3.0);
	}
	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 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	t_3 = cbrt(sqrt(2.0))
	t_4 = Float64(l * Float64(l / Om))
	tmp = 0.0
	if (t_2 <= 5e-323)
		tmp = Float64(t_3 * (exp(0.16666666666666666) ^ Float64(log(U) + log(Float64(n * Float64(t + Float64(Float64(n / Float64(Float64(Om * Om) / Float64(Float64(l * l) * U_42_))) - Float64(2.0 * t_4)))))))) ^ 3.0;
	elseif (t_2 <= Inf)
		tmp = sqrt((cbrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - fma(2.0, t_4, Float64(t_1 * Float64(U - U_42_)))))) ^ 3.0));
	else
		tmp = Float64(t_3 * (exp(0.16666666666666666) ^ Float64(log(Float64(Float64(Float64(n / Float64(Float64(Om * Om) / Float64(U - U_42_))) + Float64(2.0 / Om)) * Float64(U * Float64(-n)))) - Float64(-2.0 * log(l))))) ^ 3.0;
	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[(N[(N[(2.0 * n), $MachinePrecision] * U), $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]}, Block[{t$95$3 = N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-323], N[Power[N[(t$95$3 * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(N[Log[U], $MachinePrecision] + N[Log[N[(n * N[(t + N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[Power[N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * t$95$4 + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$3 * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(N[Log[N[(N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * (-n)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.94066e-323

   1. Initial program 23.3%

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

   3. Simplified 23.4%

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

   5. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in U around 0 44.0%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 44.0%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{\sqrt{2} \cdot 1}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}^{3} \]

      2. *-rgt-identity 44.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{2}}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}\right)}^{3} \]

      3. exp-prod 42.0%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(n \cdot \left(t - \left(-1 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U*\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) + \log U\right)}}\right)}^{3} \]

   8. Simplified 42.0%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(n \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - \frac{n}{\frac{Om \cdot Om}{U* \cdot \left(\ell \cdot \ell\right)}}\right)\right)\right) + \log U\right)}\right)}}^{3} \]

   if 4.94066e-323 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 68.7%

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

   3. Simplified 71.6%

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

      1. associate-*r* 71.6%

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

      2. fma-udef 71.6%

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

      3. associate-*l/ 64.3%

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

      4. associate-*r* 68.7%

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

      5. *-commutative 68.7%

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

      6. associate--l- 68.7%

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

      7. add-cube-cbrt 68.3%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}} \]

      8. pow3 68.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\right)}^{3}}} \]

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   5. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}}\right)}^{3}} \]

   6. Taylor expanded in l around inf 26.6%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{2} \cdot 1\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 26.6%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{\sqrt{2} \cdot 1}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3} \]

      2. *-rgt-identity 26.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{2}}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}\right)}^{3} \]

      3. exp-prod 26.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-1 \cdot \left(\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right) \cdot \left(n \cdot U\right)\right)\right) + -2 \cdot \log \left(\frac{1}{\ell}\right)\right)}}\right)}^{3} \]

   8. Simplified 26.1%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-\left(n \cdot U\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right)\right) + -2 \cdot \left(-\log \ell\right)\right)}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log U + \log \left(n \cdot \left(t + \left(\frac{n}{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}} - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{2}} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\left(\frac{n}{\frac{Om \cdot Om}{U - U*}} + \frac{2}{Om}\right) \cdot \left(U \cdot \left(-n\right)\right)\right) - -2 \cdot \log \ell\right)}\right)}^{3}\\ \end{array} \]

Alternative 5: 61.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1 \cdot \left(U - U*\right)\right)\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t_2}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t_2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2 (- t (fma 2.0 (* l (/ l Om)) (* t_1 (- U U*)))))
        (t_3
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t_2)))
     (if (<= t_3 INFINITY)
       (sqrt (pow (cbrt (* (* 2.0 (* n U)) t_2)) 3.0))
       (sqrt
        (*
         -2.0
         (*
          (- (/ 2.0 Om) (* (/ n Om) (/ (- U* U) Om)))
          (* n (* U (* l l))))))))))

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 = t - fma(2.0, (l * (l / Om)), (t_1 * (U - U_42_)));
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t_2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(pow(cbrt(((2.0 * (n * U)) * t_2)), 3.0));
	} else {
		tmp = sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	}
	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 = Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_1 * Float64(U - U_42_))))
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t_2)));
	elseif (t_3 <= Inf)
		tmp = sqrt((cbrt(Float64(Float64(2.0 * Float64(n * U)) * t_2)) ^ 3.0));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))) * Float64(n * Float64(U * Float64(l * 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[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[Power[N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   3. Simplified 25.2%

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

      1. pow1/2 25.2%

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

      2. fma-udef 25.2%

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

      3. associate-*l/ 25.2%

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

      4. associate-*r* 25.2%

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

      5. *-commutative 25.2%

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

      6. associate--l- 25.2%

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

      7. associate-*r* 25.2%

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

      8. associate-*l* 42.1%

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

   5. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/2 40.4%

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

   7. Simplified 40.4%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 68.5%

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

   3. Simplified 71.4%

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

      1. associate-*r* 71.4%

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

      2. fma-udef 71.4%

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

      3. associate-*l/ 64.1%

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

      4. associate-*r* 68.5%

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

      5. *-commutative 68.5%

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

      6. associate--l- 68.5%

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

      7. add-cube-cbrt 68.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}} \]

      8. pow3 68.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}^{3}}} \]

   5. Applied egg-rr 75.3%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)}\right)}^{3}}} \]

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in l around inf 39.4%

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

      1. unpow2 39.4%

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

      2. times-frac 45.1%

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

      3. associate-*r/ 45.1%

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

      4. metadata-eval 45.1%

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

      5. *-commutative 45.1%

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

      6. unpow2 45.1%

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

   6. Simplified 45.1%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{{\left(\sqrt[3]{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 61.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t_1\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + t_1 \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_2 0.0)
     (pow (* 2.0 (* n (* U (+ t (* t_1 -2.0))))) 0.5)
     (if (<= t_2 5e+142)
       t_2
       (if (<= t_2 INFINITY)
         (pow
          (*
           (* 2.0 (* n U))
           (+ t (* l (* l (- (* (/ n (* Om Om)) (- U* U)) (/ 2.0 Om))))))
          0.5)
         (sqrt
          (*
           -2.0
           (*
            (- (/ 2.0 Om) (* (/ n Om) (/ (- U* U) Om)))
            (* n (* U (* l l)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow((2.0 * (n * (U * (t + (t_1 * -2.0))))), 0.5);
	} else if (t_2 <= 5e+142) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.pow((2.0 * (n * (U * (t + (t_1 * -2.0))))), 0.5);
	} else if (t_2 <= 5e+142) {
		tmp = t_2;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.pow((2.0 * (n * (U * (t + (t_1 * -2.0))))), 0.5)
	elif t_2 <= 5e+142:
		tmp = t_2
	elif t_2 <= math.inf:
		tmp = math.pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(t_1 * -2.0))))) ^ 0.5;
	elseif (t_2 <= 5e+142)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) - Float64(2.0 / Om)))))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))) * Float64(n * Float64(U * Float64(l * l))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * l) / Om;
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = (2.0 * (n * (U * (t + (t_1 * -2.0))))) ^ 0.5;
	elseif (t_2 <= 5e+142)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = ((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $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$2, 0.0], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$2, 5e+142], t$95$2, If[LessEqual[t$95$2, Infinity], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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 25.2%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in l around 0 21.9%

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

      1. *-commutative 21.9%

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

      2. unpow2 21.9%

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

      3. associate-/l* 21.9%

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

      4. unpow2 21.9%

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

      5. associate-*r/ 21.9%

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

      6. metadata-eval 21.9%

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

   6. Simplified 21.9%

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

      1. pow1/2 21.9%

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

      2. associate-*l* 22.0%

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

      3. associate-/r/ 22.0%

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

   8. Applied egg-rr 22.0%

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

   9. Taylor expanded in n around 0 39.4%

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

       1. *-commutative 39.4%

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

       2. cancel-sign-sub-inv 39.4%

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

       3. metadata-eval 39.4%

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

       4. unpow2 39.4%

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

   11. Simplified 39.4%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.0000000000000001e142

   1. Initial program 96.8%

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

   if 5.0000000000000001e142 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.5%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in l around 0 31.2%

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

      1. *-commutative 31.2%

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

      2. unpow2 31.2%

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

      3. associate-/l* 32.9%

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

      4. unpow2 32.9%

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

      5. associate-*r/ 32.9%

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

      6. metadata-eval 32.9%

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

   6. Simplified 32.9%

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

      1. pow1/2 34.4%

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

      2. associate-*l* 50.9%

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

      3. associate-/r/ 50.9%

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

   8. Applied egg-rr 50.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in l around inf 39.4%

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

      1. unpow2 39.4%

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

      2. times-frac 45.1%

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

      3. associate-*r/ 45.1%

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

      4. metadata-eval 45.1%

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

      5. *-commutative 45.1%

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

      6. unpow2 45.1%

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

   6. Simplified 45.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7: 61.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot t_1\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + t_1 \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t + 2 \cdot \frac{-1}{\frac{Om}{\ell \cdot \ell}}\right) + t_2\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))))
   (if (<= t_4 0.0)
     (pow (* 2.0 (* n (* U (+ t (* t_1 -2.0))))) 0.5)
     (if (<= t_4 5e+142)
       (sqrt (* t_3 (+ (+ t (* 2.0 (/ -1.0 (/ Om (* l l))))) t_2)))
       (if (<= t_4 INFINITY)
         (pow
          (*
           (* 2.0 (* n U))
           (+ t (* l (* l (- (* (/ n (* Om Om)) (- U* U)) (/ 2.0 Om))))))
          0.5)
         (sqrt
          (*
           -2.0
           (*
            (- (/ 2.0 Om) (* (/ n Om) (/ (- U* U) Om)))
            (* n (* U (* l l)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = pow((2.0 * (n * (U * (t + (t_1 * -2.0))))), 0.5);
	} else if (t_4 <= 5e+142) {
		tmp = sqrt((t_3 * ((t + (2.0 * (-1.0 / (Om / (l * l))))) + t_2)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.pow((2.0 * (n * (U * (t + (t_1 * -2.0))))), 0.5);
	} else if (t_4 <= 5e+142) {
		tmp = Math.sqrt((t_3 * ((t + (2.0 * (-1.0 / (Om / (l * l))))) + t_2)));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.pow((2.0 * (n * (U * (t + (t_1 * -2.0))))), 0.5)
	elif t_4 <= 5e+142:
		tmp = math.sqrt((t_3 * ((t + (2.0 * (-1.0 / (Om / (l * l))))) + t_2)))
	elif t_4 <= math.inf:
		tmp = math.pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(t_1 * -2.0))))) ^ 0.5;
	elseif (t_4 <= 5e+142)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Om / Float64(l * l))))) + t_2)));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) - Float64(2.0 / Om)))))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))) * Float64(n * Float64(U * Float64(l * l))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * l) / Om;
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = (2.0 * (n * (U * (t + (t_1 * -2.0))))) ^ 0.5;
	elseif (t_4 <= 5e+142)
		tmp = sqrt((t_3 * ((t + (2.0 * (-1.0 / (Om / (l * l))))) + t_2)));
	elseif (t_4 <= Inf)
		tmp = ((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$4, 5e+142], N[Sqrt[N[(t$95$3 * N[(N[(t + N[(2.0 * N[(-1.0 / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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 25.2%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in l around 0 21.9%

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

      1. *-commutative 21.9%

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

      2. unpow2 21.9%

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

      3. associate-/l* 21.9%

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

      4. unpow2 21.9%

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

      5. associate-*r/ 21.9%

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

      6. metadata-eval 21.9%

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

   6. Simplified 21.9%

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

      1. pow1/2 21.9%

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

      2. associate-*l* 22.0%

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

      3. associate-/r/ 22.0%

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

   8. Applied egg-rr 22.0%

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

   9. Taylor expanded in n around 0 39.4%

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

       1. *-commutative 39.4%

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

       2. cancel-sign-sub-inv 39.4%

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

       3. metadata-eval 39.4%

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

       4. unpow2 39.4%

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

   11. Simplified 39.4%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.0000000000000001e142

   1. Initial program 96.8%

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

      1. clear-num 96.8%

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

      2. inv-pow 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. unpow-1 96.8%

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

   5. Simplified 96.8%

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

   if 5.0000000000000001e142 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.5%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in l around 0 31.2%

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

      1. *-commutative 31.2%

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

      2. unpow2 31.2%

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

      3. associate-/l* 32.9%

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

      4. unpow2 32.9%

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

      5. associate-*r/ 32.9%

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

      6. metadata-eval 32.9%

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

   6. Simplified 32.9%

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

      1. pow1/2 34.4%

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

      2. associate-*l* 50.9%

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

      3. associate-/r/ 50.9%

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

   8. Applied egg-rr 50.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in l around inf 39.4%

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

      1. unpow2 39.4%

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

      2. times-frac 45.1%

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

      3. associate-*r/ 45.1%

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

      4. metadata-eval 45.1%

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

      5. *-commutative 45.1%

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

      6. unpow2 45.1%

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

   6. Simplified 45.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + 2 \cdot \frac{-1}{\frac{Om}{\ell \cdot \ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8: 62.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := t_2 \cdot \left(U* - U\right)\\ t_4 := \sqrt{t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_3\right)}\\ \mathbf{if}\;t_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_2 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t + 2 \cdot \frac{-1}{\frac{Om}{\ell \cdot \ell}}\right) + t_3\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3 (* t_2 (- U* U)))
        (t_4 (sqrt (* t_1 (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_4 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (fma 2.0 (* l (/ l Om)) (* t_2 (- U U*)))))))
     (if (<= t_4 5e+142)
       (sqrt (* t_1 (+ (+ t (* 2.0 (/ -1.0 (/ Om (* l l))))) t_3)))
       (if (<= t_4 INFINITY)
         (pow
          (*
           (* 2.0 (* n U))
           (+ t (* l (* l (- (* (/ n (* Om Om)) (- U* U)) (/ 2.0 Om))))))
          0.5)
         (sqrt
          (*
           -2.0
           (*
            (- (/ 2.0 Om) (* (/ n Om) (/ (- U* U) Om)))
            (* n (* U (* l l)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = n * pow((l / Om), 2.0);
	double t_3 = t_2 * (U_42_ - U);
	double t_4 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - fma(2.0, (l * (l / Om)), (t_2 * (U - U_42_))))));
	} else if (t_4 <= 5e+142) {
		tmp = sqrt((t_1 * ((t + (2.0 * (-1.0 / (Om / (l * l))))) + t_3)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = pow(((2.0 * (n * U)) * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (((2.0 / Om) - ((n / Om) * ((U_42_ - U) / Om))) * (n * (U * (l * l))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = Float64(t_2 * Float64(U_42_ - U))
	t_4 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(t_2 * Float64(U - U_42_)))))));
	elseif (t_4 <= 5e+142)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t + Float64(2.0 * Float64(-1.0 / Float64(Om / Float64(l * l))))) + t_3)));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) - Float64(2.0 / Om)))))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(2.0 / Om) - Float64(Float64(n / Om) * Float64(Float64(U_42_ - U) / Om))) * Float64(n * Float64(U * Float64(l * 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+142], N[Sqrt[N[(t$95$1 * N[(N[(t + N[(2.0 * N[(-1.0 / N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(l * N[(l * N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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 25.2%

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

   3. Simplified 25.2%

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

      1. pow1/2 25.2%

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

      2. fma-udef 25.2%

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

      3. associate-*l/ 25.2%

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

      4. associate-*r* 25.2%

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

      5. *-commutative 25.2%

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

      6. associate--l- 25.2%

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

      7. associate-*r* 25.2%

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

      8. associate-*l* 42.1%

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

   5. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/2 40.4%

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

   7. Simplified 40.4%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.0000000000000001e142

   1. Initial program 96.8%

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

      1. clear-num 96.8%

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

      2. inv-pow 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. unpow-1 96.8%

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

   5. Simplified 96.8%

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

   if 5.0000000000000001e142 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.5%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in l around 0 31.2%

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

      1. *-commutative 31.2%

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

      2. unpow2 31.2%

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

      3. associate-/l* 32.9%

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

      4. unpow2 32.9%

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

      5. associate-*r/ 32.9%

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

      6. metadata-eval 32.9%

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

   6. Simplified 32.9%

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

      1. pow1/2 34.4%

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

      2. associate-*l* 50.9%

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

      3. associate-/r/ 50.9%

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

   8. Applied egg-rr 50.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.6%

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

   4. Taylor expanded in l around inf 39.4%

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

      1. unpow2 39.4%

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

      2. times-frac 45.1%

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

      3. associate-*r/ 45.1%

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

      4. metadata-eval 45.1%

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

      5. *-commutative 45.1%

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

      6. unpow2 45.1%

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

   6. Simplified 45.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + 2 \cdot \frac{-1}{\frac{Om}{\ell \cdot \ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\frac{2}{Om} - \frac{n}{Om} \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}\\ \end{array} \]

Alternative 9: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := \sqrt{t_1 \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{if}\;\ell \leq 6 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om} + \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(t_1 \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (+ t (* (/ (* l l) Om) -2.0))
            (* n (* (pow (/ l Om) 2.0) (- U* U))))))))
   (if (<= l 6e-107)
     t_2
     (if (<= l 5.4e+38)
       (sqrt
        (*
         2.0
         (*
          U
          (*
           n
           (+
            (* (/ n Om) (/ (* (* l l) U*) Om))
            (fma l (* (/ l Om) -2.0) t))))))
       (if (<= l 9.2e+42)
         t_2
         (pow
          (* t_1 (+ t (* l (* l (- (* (/ n (* Om Om)) (- U* U)) (/ 2.0 Om))))))
          0.5))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = sqrt((t_1 * ((t + (((l * l) / Om) * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - U))))));
	double tmp;
	if (l <= 6e-107) {
		tmp = t_2;
	} else if (l <= 5.4e+38) {
		tmp = sqrt((2.0 * (U * (n * (((n / Om) * (((l * l) * U_42_) / Om)) + fma(l, ((l / Om) * -2.0), t))))));
	} else if (l <= 9.2e+42) {
		tmp = t_2;
	} else {
		tmp = pow((t_1 * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U))))))
	tmp = 0.0
	if (l <= 6e-107)
		tmp = t_2;
	elseif (l <= 5.4e+38)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(Float64(Float64(n / Om) * Float64(Float64(Float64(l * l) * U_42_) / Om)) + fma(l, Float64(Float64(l / Om) * -2.0), t))))));
	elseif (l <= 9.2e+42)
		tmp = t_2;
	else
		tmp = Float64(t_1 * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) - Float64(2.0 / Om)))))) ^ 0.5;
	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[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 6e-107], t$95$2, If[LessEqual[l, 5.4e+38], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(l * N[(N[(l / Om), $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.2e+42], t$95$2, N[Power[N[(t$95$1 * N[(t + N[(l * N[(l * N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.9999999999999994e-107 or 5.39999999999999992e38 < l < 9.2e42

   1. Initial program 53.5%

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

   3. Simplified 51.3%

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

   if 5.9999999999999994e-107 < l < 5.39999999999999992e38

   1. Initial program 42.2%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in U around 0 42.2%

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

      1. associate-*r* 52.8%

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

   6. Simplified 58.2%

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

   if 9.2e42 < l

   1. Initial program 37.2%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in l around 0 47.7%

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

      1. *-commutative 47.7%

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

      2. unpow2 47.7%

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

      3. associate-/l* 47.7%

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

      4. unpow2 47.7%

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

      5. associate-*r/ 47.7%

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

      6. metadata-eval 47.7%

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

   6. Simplified 47.7%

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

      1. pow1/2 48.0%

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

      2. associate-*l* 53.5%

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

      3. associate-/r/ 53.4%

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

   8. Applied egg-rr 53.4%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(\frac{n}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om} + \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot -2, t\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 10: 53.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(t_1 \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 4.7e-166)
     (sqrt (* t t_1))
     (pow
      (* t_1 (+ t (* l (* l (- (* (/ n (* Om Om)) (- U* U)) (/ 2.0 Om))))))
      0.5))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 4.7e-166) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = pow((t_1 * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (l <= 4.7d-166) then
        tmp = sqrt((t * t_1))
    else
        tmp = (t_1 * (t + (l * (l * (((n / (om * om)) * (u_42 - u)) - (2.0d0 / om)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 4.7e-166) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.pow((t_1 * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if l <= 4.7e-166:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.pow((t_1 * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 4.7e-166)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = Float64(t_1 * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(U_42_ - U)) - Float64(2.0 / Om)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 4.7e-166)
		tmp = sqrt((t * t_1));
	else
		tmp = (t_1 * (t + (l * (l * (((n / (Om * Om)) * (U_42_ - U)) - (2.0 / Om)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.7e-166], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$1 * N[(t + N[(l * N[(l * N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.70000000000000014e-166

   1. Initial program 52.4%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in t around inf 39.7%

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

   if 4.70000000000000014e-166 < l

   1. Initial program 43.6%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in l around 0 45.4%

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

      1. *-commutative 45.4%

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

      2. unpow2 45.4%

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

      3. associate-/l* 45.5%

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

      4. unpow2 45.5%

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

      5. associate-*r/ 45.5%

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

      6. metadata-eval 45.5%

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

   6. Simplified 45.5%

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

      1. pow1/2 45.8%

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

      2. associate-*l* 49.9%

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

      3. associate-/r/ 49.9%

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

   8. Applied egg-rr 49.9%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n}{Om \cdot Om} \cdot \left(U* - U\right) - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 11: 53.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{t \cdot t_1}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 1.5e-165)
     (sqrt (* t t_1))
     (if (<= l 1.7e+149)
       (sqrt (* t_1 (+ t (* (* l l) (- (/ n (/ (* Om Om) U*)) (/ 2.0 Om))))))
       (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ l (/ Om l)))))) 0.5)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 1.5e-165) {
		tmp = sqrt((t * t_1));
	} else if (l <= 1.7e+149) {
		tmp = sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	} else {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (l <= 1.5d-165) then
        tmp = sqrt((t * t_1))
    else if (l <= 1.7d+149) then
        tmp = sqrt((t_1 * (t + ((l * l) * ((n / ((om * om) / u_42)) - (2.0d0 / om))))))
    else
        tmp = ((2.0d0 * n) * (u * (t - (2.0d0 * (l / (om / l)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 1.5e-165) {
		tmp = Math.sqrt((t * t_1));
	} else if (l <= 1.7e+149) {
		tmp = Math.sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if l <= 1.5e-165:
		tmp = math.sqrt((t * t_1))
	elif l <= 1.7e+149:
		tmp = math.sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))))
	else:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 1.5e-165)
		tmp = sqrt(Float64(t * t_1));
	elseif (l <= 1.7e+149)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l * l) * Float64(Float64(n / Float64(Float64(Om * Om) / U_42_)) - Float64(2.0 / Om))))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 1.5e-165)
		tmp = sqrt((t * t_1));
	elseif (l <= 1.7e+149)
		tmp = sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	else
		tmp = ((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.5e-165], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.7e+149], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.49999999999999989e-165

   1. Initial program 52.4%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in t around inf 39.7%

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

   if 1.49999999999999989e-165 < l < 1.6999999999999999e149

   1. Initial program 55.8%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in l around 0 50.5%

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

      1. *-commutative 50.5%

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

      2. unpow2 50.5%

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

      3. associate-/l* 50.8%

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

      4. unpow2 50.8%

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

      5. associate-*r/ 50.8%

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

      6. metadata-eval 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in U around 0 50.2%

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

      1. unpow2 50.2%

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

      2. associate-*r/ 50.2%

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

      3. metadata-eval 50.2%

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

      4. +-commutative 50.2%

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

      5. mul-1-neg 50.2%

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

      6. unsub-neg 50.2%

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

      7. associate-/l* 50.2%

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

      8. unpow2 50.2%

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

   9. Simplified 50.2%

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

   if 1.6999999999999999e149 < l

   1. Initial program 24.1%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in l around 0 37.1%

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

      1. *-commutative 37.1%

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

      2. unpow2 37.1%

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

      3. associate-/l* 37.1%

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

      4. unpow2 37.1%

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

      5. associate-*r/ 37.1%

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

      6. metadata-eval 37.1%

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

   6. Simplified 37.1%

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

      1. pow1/2 37.5%

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

      2. associate-*l* 46.0%

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

      3. associate-/r/ 46.0%

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

   8. Applied egg-rr 46.0%

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

   9. Taylor expanded in n around 0 38.2%

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

       1. associate-*r* 38.2%

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

       2. *-commutative 38.2%

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

       3. *-commutative 38.2%

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

       4. unpow2 38.2%

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

       5. associate-/l* 46.6%

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

       6. *-commutative 46.6%

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

   11. Simplified 46.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 12: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;U \leq -1.05 \cdot 10^{-157}:\\ \;\;\;\;{\left(t_1 \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 0.0012:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= U -1.05e-157)
     (pow (* t_1 (+ t (* l (* l (- (/ (* n U*) (* Om Om)) (/ 2.0 Om)))))) 0.5)
     (if (<= U 0.0012)
       (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ l (/ Om l)))))) 0.5)
       (sqrt
        (* t_1 (+ t (* (* l l) (- (/ n (/ (* Om Om) U*)) (/ 2.0 Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (U <= -1.05e-157) {
		tmp = pow((t_1 * (t + (l * (l * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))))), 0.5);
	} else if (U <= 0.0012) {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5);
	} else {
		tmp = sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (u <= (-1.05d-157)) then
        tmp = (t_1 * (t + (l * (l * (((n * u_42) / (om * om)) - (2.0d0 / om)))))) ** 0.5d0
    else if (u <= 0.0012d0) then
        tmp = ((2.0d0 * n) * (u * (t - (2.0d0 * (l / (om / l)))))) ** 0.5d0
    else
        tmp = sqrt((t_1 * (t + ((l * l) * ((n / ((om * om) / u_42)) - (2.0d0 / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (U <= -1.05e-157) {
		tmp = Math.pow((t_1 * (t + (l * (l * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))))), 0.5);
	} else if (U <= 0.0012) {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5);
	} else {
		tmp = Math.sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if U <= -1.05e-157:
		tmp = math.pow((t_1 * (t + (l * (l * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))))), 0.5)
	elif U <= 0.0012:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5)
	else:
		tmp = math.sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (U <= -1.05e-157)
		tmp = Float64(t_1 * Float64(t + Float64(l * Float64(l * Float64(Float64(Float64(n * U_42_) / Float64(Om * Om)) - Float64(2.0 / Om)))))) ^ 0.5;
	elseif (U <= 0.0012)
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))))) ^ 0.5;
	else
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l * l) * Float64(Float64(n / Float64(Float64(Om * Om) / U_42_)) - Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (U <= -1.05e-157)
		tmp = (t_1 * (t + (l * (l * (((n * U_42_) / (Om * Om)) - (2.0 / Om)))))) ^ 0.5;
	elseif (U <= 0.0012)
		tmp = ((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))) ^ 0.5;
	else
		tmp = sqrt((t_1 * (t + ((l * l) * ((n / ((Om * Om) / U_42_)) - (2.0 / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.05e-157], N[Power[N[(t$95$1 * N[(t + N[(l * N[(l * N[(N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 0.0012], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l * l), $MachinePrecision] * N[(N[(n / N[(N[(Om * Om), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if U < -1.05e-157

   1. Initial program 54.7%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in l around 0 56.0%

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

      1. *-commutative 56.0%

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

      2. unpow2 56.0%

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

      3. associate-/l* 54.9%

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

      4. unpow2 54.9%

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

      5. associate-*r/ 54.9%

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

      6. metadata-eval 54.9%

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

   6. Simplified 54.9%

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

      1. pow1/2 55.0%

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

      2. associate-*l* 62.3%

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

      3. associate-/r/ 63.4%

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

   8. Applied egg-rr 63.4%

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

   9. Taylor expanded in U around 0 64.5%

      \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \ell \cdot \left(\ell \cdot \left(\color{blue}{-1 \cdot \frac{n \cdot U*}{{Om}^{2}}} + \frac{2}{Om}\right)\right)\right)\right)}^{0.5} \]
   10. Step-by-step derivation

       1. associate-*r/ 64.5%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{-1 \cdot \left(n \cdot U*\right)}{{Om}^{2}}} + \frac{2}{Om}\right)\right)\right)\right)}^{0.5} \]

       2. mul-1-neg 64.5%

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

       3. unpow2 64.5%

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

   11. Simplified 64.5%

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

   if -1.05e-157 < U < 0.00119999999999999989

   1. Initial program 43.7%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in l around 0 38.5%

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

      1. *-commutative 38.5%

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

      2. unpow2 38.5%

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

      3. associate-/l* 39.9%

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

      4. unpow2 39.9%

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

      5. associate-*r/ 39.9%

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

      6. metadata-eval 39.9%

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

   6. Simplified 39.9%

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

      1. pow1/2 40.0%

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

      2. associate-*l* 45.6%

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

      3. associate-/r/ 44.9%

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

   8. Applied egg-rr 44.9%

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

   9. Taylor expanded in n around 0 42.9%

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

       1. associate-*r* 43.6%

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

       2. *-commutative 43.6%

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

       3. *-commutative 43.6%

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

       4. unpow2 43.6%

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

       5. associate-/l* 50.3%

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

       6. *-commutative 50.3%

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

   11. Simplified 50.3%

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

   if 0.00119999999999999989 < U

   1. Initial program 61.4%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in l around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. unpow2 60.8%

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

      3. associate-/l* 68.3%

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

      4. unpow2 68.3%

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

      5. associate-*r/ 68.3%

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

      6. metadata-eval 68.3%

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

   6. Simplified 68.3%

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

   7. Taylor expanded in U around 0 65.1%

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

      1. unpow2 65.1%

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

      2. associate-*r/ 65.1%

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

      3. metadata-eval 65.1%

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

      4. +-commutative 65.1%

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

      5. mul-1-neg 65.1%

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

      6. unsub-neg 65.1%

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

      7. associate-/l* 71.8%

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

      8. unpow2 71.8%

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

   9. Simplified 71.8%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.05 \cdot 10^{-157}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \ell \cdot \left(\ell \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \frac{2}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 0.0012:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\ell \cdot \ell\right) \cdot \left(\frac{n}{\frac{Om \cdot Om}{U*}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]

Alternative 13: 51.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.8 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 10^{-95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.8e+122)
   (sqrt (* (* 2.0 (* n U)) (- t (* 2.0 (* l (/ l Om))))))
   (if (<= Om 1e-95)
     (pow (* 2.0 (* n (* U (+ t (* (/ (* l l) Om) -2.0))))) 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 (Om <= -1.8e+122) {
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 * (l * (l / Om))))));
	} else if (Om <= 1e-95) {
		tmp = pow((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))), 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 (om <= (-1.8d+122)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (2.0d0 * (l * (l / om))))))
    else if (om <= 1d-95) then
        tmp = (2.0d0 * (n * (u * (t + (((l * l) / om) * (-2.0d0)))))) ** 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 (Om <= -1.8e+122) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - (2.0 * (l * (l / Om))))));
	} else if (Om <= 1e-95) {
		tmp = Math.pow((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))), 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 Om <= -1.8e+122:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - (2.0 * (l * (l / Om))))))
	elif Om <= 1e-95:
		tmp = math.pow((2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))), 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 (Om <= -1.8e+122)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))));
	elseif (Om <= 1e-95)
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * 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 (Om <= -1.8e+122)
		tmp = sqrt(((2.0 * (n * U)) * (t - (2.0 * (l * (l / Om))))));
	elseif (Om <= 1e-95)
		tmp = (2.0 * (n * (U * (t + (((l * l) / Om) * -2.0))))) ^ 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[Om, -1.8e+122], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1e-95], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.8000000000000001e122

   1. Initial program 48.3%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in Om around inf 44.5%

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

      1. unpow2 44.5%

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

      2. associate-*r/ 59.2%

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

   6. Simplified 59.2%

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

   if -1.8000000000000001e122 < Om < 9.99999999999999989e-96

   1. Initial program 43.0%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in l around 0 41.6%

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

      1. *-commutative 41.6%

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

      2. unpow2 41.6%

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

      3. associate-/l* 41.7%

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

      4. unpow2 41.7%

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

      5. associate-*r/ 41.7%

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

      6. metadata-eval 41.7%

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

   6. Simplified 41.7%

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

      1. pow1/2 41.9%

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

      2. associate-*l* 43.1%

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

      3. associate-/r/ 42.3%

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

   8. Applied egg-rr 42.3%

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

   9. Taylor expanded in n around 0 43.8%

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

       1. *-commutative 43.8%

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

       2. cancel-sign-sub-inv 43.8%

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

       3. metadata-eval 43.8%

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

       4. unpow2 43.8%

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

   11. Simplified 43.8%

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

   if 9.99999999999999989e-96 < Om

   1. Initial program 59.4%

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

      1. clear-num 59.4%

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

      2. inv-pow 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. unpow-1 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in Om around inf 53.5%

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

      1. metadata-eval 53.5%

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

      2. cancel-sign-sub-inv 53.5%

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

      3. unpow2 53.5%

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

      4. associate-*r/ 59.5%

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

      5. associate-*r* 59.5%

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

   8. Simplified 59.5%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.8 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 10^{-95}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)}\\ \end{array} \]

Alternative 14: 51.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.5e-85)
   (sqrt (* t (* 2.0 (* n U))))
   (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ l (/ Om l)))))) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.5e-85) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.5d-85) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = ((2.0d0 * n) * (u * (t - (2.0d0 * (l / (om / l)))))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.5e-85) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.5e-85:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.5e-85)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.5e-85)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = ((2.0 * n) * (U * (t - (2.0 * (l / (Om / l)))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.5e-85], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.50000000000000011e-85

   1. Initial program 52.2%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in t around inf 39.3%

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

   if 1.50000000000000011e-85 < l

   1. Initial program 41.6%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in l around 0 48.1%

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

      1. *-commutative 48.1%

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

      2. unpow2 48.1%

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

      3. associate-/l* 48.1%

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

      4. unpow2 48.1%

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

      5. associate-*r/ 48.1%

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

      6. metadata-eval 48.1%

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

   6. Simplified 48.1%

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

      1. pow1/2 48.3%

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

      2. associate-*l* 52.5%

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

      3. associate-/r/ 52.4%

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

   8. Applied egg-rr 52.4%

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

   9. Taylor expanded in n around 0 43.2%

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

       1. associate-*r* 44.7%

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

       2. *-commutative 44.7%

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

       3. *-commutative 44.7%

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

       4. unpow2 44.7%

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

       5. associate-/l* 48.8%

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

       6. *-commutative 48.8%

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

   11. Simplified 48.8%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 15: 45.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 51.8%

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

4. Taylor expanded in Om around inf 41.3%

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

   1. unpow2 41.3%

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

   2. associate-*r/ 46.2%

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

6. Simplified 46.2%

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

7. Final simplification 46.2%

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

Alternative 16: 45.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. clear-num 49.4%

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

   2. inv-pow 49.4%

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

3. Applied egg-rr 49.4%

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

   1. unpow-1 49.4%

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

5. Simplified 49.4%

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

6. Taylor expanded in Om around inf 41.3%

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

   1. metadata-eval 41.3%

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

   2. cancel-sign-sub-inv 41.3%

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

   3. unpow2 41.3%

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

   4. associate-*r/ 46.2%

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

   5. associate-*r* 46.2%

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

8. Simplified 46.2%

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

9. Final simplification 46.2%

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

Alternative 17: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.1e-106)
   (sqrt (* t (* 2.0 (* n U))))
   (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_) {
	double tmp;
	if (l <= 4.1e-106) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.1d-106) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.1e-106) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.1e-106:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.1e-106)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.1e-106)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.1e-106], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.0999999999999999e-106

   1. Initial program 52.7%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in t around inf 39.5%

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

   if 4.0999999999999999e-106 < l

   1. Initial program 41.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in t around inf 13.3%

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

      1. pow1/2 14.6%

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

      2. associate-*r* 17.6%

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

   6. Applied egg-rr 17.6%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 18: 35.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.1e-62) (sqrt (* t (* 2.0 (* n U)))) (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_) {
	double tmp;
	if (l <= 2.1e-62) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.1d-62) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.1e-62) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.1e-62:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.1e-62)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.1e-62)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.1e-62], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.0999999999999999e-62

   1. Initial program 52.5%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around inf 39.1%

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

   if 2.0999999999999999e-62 < l

   1. Initial program 40.7%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in t around inf 12.4%

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

      1. associate-*r* 13.9%

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

   6. Simplified 13.9%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 19: 33.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 51.8%

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

4. Taylor expanded in t around inf 29.9%

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

5. Final simplification 29.9%

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

Alternative 20: 34.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 51.8%

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

4. Taylor expanded in t around inf 29.9%

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

   1. associate-*r* 30.7%

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

6. Simplified 30.7%

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

7. Final simplification 30.7%

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

Reproduce

herbie shell --seed 2023272 
(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*))))))