Toniolo and Linder, Equation (13)

Percentage Accurate: 50.4% → 62.3%

Time: 25.0s

Alternatives: 18

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{n \cdot 2}\\ \mathbf{if}\;n \leq -7 \cdot 10^{-35}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.14 \cdot 10^{+207} \lor \neg \left(n \leq 1.2 \cdot 10^{+252}\right):\\ \;\;\;\;t_1 \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{\frac{U \cdot U*}{Om}}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* n 2.0))))
   (if (<= n -7e-35)
     (pow
      (*
       2.0
       (+
        (* n (* t U))
        (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
      0.5)
     (if (<= n 2.3e-307)
       (sqrt
        (*
         2.0
         (*
          U
          (+
           (* n t)
           (* (/ (fma (* l (/ n Om)) (- U* U) (* l -2.0)) Om) (* n l))))))
       (if (<= n 1.05e+101)
         (*
          t_1
          (sqrt
           (* U (+ t (* (/ l Om) (fma l -2.0 (* (* n (- U* U)) (/ l Om))))))))
         (if (or (<= n 1.14e+207) (not (<= n 1.2e+252)))
           (*
            t_1
            (sqrt
             (*
              U
              (+
               (fma -2.0 (* l (/ l Om)) t)
               (* n (* (- U* U) (pow (/ l Om) 2.0)))))))
           (* (* l (sqrt 2.0)) (* n (sqrt (/ (/ (* U U*) Om) Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((n * 2.0));
	double tmp;
	if (n <= -7e-35) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 2.3e-307) {
		tmp = sqrt((2.0 * (U * ((n * t) + ((fma((l * (n / Om)), (U_42_ - U), (l * -2.0)) / Om) * (n * l))))));
	} else if (n <= 1.05e+101) {
		tmp = t_1 * sqrt((U * (t + ((l / Om) * fma(l, -2.0, ((n * (U_42_ - U)) * (l / Om)))))));
	} else if ((n <= 1.14e+207) || !(n <= 1.2e+252)) {
		tmp = t_1 * sqrt((U * (fma(-2.0, (l * (l / Om)), t) + (n * ((U_42_ - U) * pow((l / Om), 2.0))))));
	} else {
		tmp = (l * sqrt(2.0)) * (n * sqrt((((U * U_42_) / Om) / Om)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(n * 2.0))
	tmp = 0.0
	if (n <= -7e-35)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 2.3e-307)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(Float64(fma(Float64(l * Float64(n / Om)), Float64(U_42_ - U), Float64(l * -2.0)) / Om) * Float64(n * l))))));
	elseif (n <= 1.05e+101)
		tmp = Float64(t_1 * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(U_42_ - U)) * Float64(l / Om))))))));
	elseif ((n <= 1.14e+207) || !(n <= 1.2e+252))
		tmp = Float64(t_1 * sqrt(Float64(U * Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) + Float64(n * Float64(Float64(U_42_ - U) * (Float64(l / Om) ^ 2.0)))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(n * sqrt(Float64(Float64(Float64(U * U_42_) / Om) / Om))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -7e-35], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 2.3e-307], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.05e+101], N[(t$95$1 * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 1.14e+207], N[Not[LessEqual[n, 1.2e+252]], $MachinePrecision]], N[(t$95$1 * N[Sqrt[N[(U * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] + N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[N[(N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -6.99999999999999992e-35

   1. Initial program 43.9%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in t around inf 60.0%

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

      1. pow1/2 60.1%

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

      2. distribute-lft-out 60.1%

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

      3. associate-/l* 60.2%

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

      4. associate-/l* 60.2%

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

      5. *-commutative 60.2%

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

      6. *-commutative 60.2%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in U around 0 60.3%

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

   if -6.99999999999999992e-35 < n < 2.2999999999999999e-307

   1. Initial program 43.1%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in t around inf 59.3%

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

      1. pow1/2 59.7%

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

      2. distribute-lft-out 59.7%

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

      3. associate-/l* 58.3%

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

      4. associate-/l* 56.9%

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

      5. *-commutative 56.9%

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

      6. *-commutative 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. *-un-lft-identity 56.9%

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

      2. unpow1/2 56.5%

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

      3. fma-def 56.5%

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

      4. +-commutative 56.5%

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

      5. associate-/r/ 57.8%

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

      6. *-commutative 57.8%

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

      7. associate-*r* 56.4%

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

   8. Applied egg-rr 56.4%

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

      1. *-lft-identity 56.4%

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

      2. fma-udef 56.4%

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

      3. associate-*r* 60.5%

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

      4. associate-/r/ 56.4%

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

      5. associate-*r* 57.2%

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

      6. distribute-rgt-out 57.1%

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

   10. Simplified 70.0%

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

   if 2.2999999999999999e-307 < n < 1.05e101

   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 61.1%

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

      1. sqrt-prod 71.6%

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

   5. Applied egg-rr 71.6%

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

      1. *-commutative 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   if 1.05e101 < n < 1.14e207 or 1.2e252 < n

   1. Initial program 54.2%

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

      1. associate-*l* 54.2%

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

      2. sub-neg 54.2%

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

      3. associate-+l- 54.2%

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

      4. sub-neg 54.2%

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

      5. associate-/l* 54.2%

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

      6. remove-double-neg 54.2%

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

      7. associate-*l* 54.4%

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

   3. Simplified 54.4%

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

      1. sqrt-prod 96.5%

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

      2. fma-def 96.5%

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

      3. associate-/l* 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. fma-udef 96.5%

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

      2. unpow2 96.5%

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

      3. associate--r+ 96.5%

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

      4. cancel-sign-sub-inv 96.5%

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

      5. metadata-eval 96.5%

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

      6. +-commutative 96.5%

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

      7. unpow2 96.5%

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

      8. fma-def 96.5%

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

      9. associate-*r/ 96.5%

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

      10. *-commutative 96.5%

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

   7. Simplified 96.5%

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

   if 1.14e207 < n < 1.2e252

   1. Initial program 42.8%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in t around inf 43.1%

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

   5. Taylor expanded in l around inf 41.8%

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

   6. Taylor expanded in n around inf 51.2%

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

   7. Taylor expanded in U* around inf 51.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-35}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.14 \cdot 10^{+207} \lor \neg \left(n \leq 1.2 \cdot 10^{+252}\right):\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{\frac{U \cdot U*}{Om}}{Om}}\right)\\ \end{array} \]

Alternative 2: 68.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{t_1}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(\left(U \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \frac{1}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* U (* n 2.0))
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (- U* U) (* n (pow (/ l Om) 2.0))))))
        (t_2
         (sqrt
          (*
           2.0
           (*
            U
            (+
             (* n t)
             (* (/ (fma (* l (/ n Om)) (- U* U) (* l -2.0)) Om) (* n l))))))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 5e+276)
       (sqrt t_1)
       (if (<= t_1 INFINITY)
         t_2
         (*
          (* l (sqrt 2.0))
          (sqrt
           (* n (* (* U (+ -2.0 (* (/ n Om) (- U* U)))) (/ 1.0 Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) + ((U_42_ - U) * (n * pow((l / Om), 2.0))));
	double t_2 = sqrt((2.0 * (U * ((n * t) + ((fma((l * (n / Om)), (U_42_ - U), (l * -2.0)) / Om) * (n * l))))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+276) {
		tmp = sqrt(t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * ((U * (-2.0 + ((n / Om) * (U_42_ - U)))) * (1.0 / Om))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(U_42_ - U) * Float64(n * (Float64(l / Om) ^ 2.0)))))
	t_2 = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(Float64(fma(Float64(l * Float64(n / Om)), Float64(U_42_ - U), Float64(l * -2.0)) / Om) * Float64(n * l))))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+276)
		tmp = sqrt(t_1);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(n * Float64(Float64(U * Float64(-2.0 + Float64(Float64(n / Om) * Float64(U_42_ - U)))) * Float64(1.0 / Om)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+276], N[Sqrt[t$95$1], $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(n * N[(N[(U * N[(-2.0 + N[(N[(n / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 0.0 or 5.00000000000000001e276 < (*.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 22.4%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in t around inf 39.1%

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

      1. pow1/2 39.4%

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

      2. distribute-lft-out 39.4%

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

      3. associate-/l* 38.7%

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

      4. associate-/l* 38.6%

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

      5. *-commutative 38.6%

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

      6. *-commutative 38.6%

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

   6. Applied egg-rr 38.6%

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

      1. *-un-lft-identity 38.6%

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

      2. unpow1/2 38.3%

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

      3. fma-def 38.3%

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

      4. +-commutative 38.3%

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

      5. associate-/r/ 38.4%

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

      6. *-commutative 38.4%

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

      7. associate-*r* 38.4%

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

   8. Applied egg-rr 38.4%

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

      1. *-lft-identity 38.4%

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

      2. fma-udef 38.4%

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

      3. associate-*r* 38.1%

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

      4. associate-/r/ 38.2%

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

      5. associate-*r* 39.0%

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

      6. distribute-rgt-out 39.1%

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

   10. Simplified 44.6%

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

   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*)))) < 5.00000000000000001e276

   1. Initial program 98.9%

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

   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 50.8%

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

   4. Taylor expanded in t around inf 59.0%

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

   5. Taylor expanded in l around inf 36.9%

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

      1. div-inv 36.8%

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

      2. *-commutative 36.8%

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

      3. sub-neg 36.8%

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

      4. associate-/l* 32.0%

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

      5. metadata-eval 32.0%

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

   7. Applied egg-rr 32.0%

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

      1. associate-*l* 33.8%

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

      2. +-commutative 33.8%

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

      3. associate-/r/ 38.5%

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

   9. Simplified 38.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(\left(U \cdot \left(-2 + \frac{n}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \frac{1}{Om}\right)}\\ \end{array} \]

Alternative 3: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.9e-36)
   (pow
    (*
     2.0
     (+
      (* n (* t U))
      (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
    0.5)
   (if (<= n 5e-309)
     (sqrt
      (*
       2.0
       (*
        U
        (+
         (* n t)
         (* (/ (fma (* l (/ n Om)) (- U* U) (* l -2.0)) Om) (* n l))))))
     (*
      (sqrt (* n 2.0))
      (sqrt
       (* U (+ t (* (/ l Om) (fma l -2.0 (* (* n (- U* U)) (/ l Om)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.9e-36) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 5e-309) {
		tmp = sqrt((2.0 * (U * ((n * t) + ((fma((l * (n / Om)), (U_42_ - U), (l * -2.0)) / Om) * (n * l))))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t + ((l / Om) * fma(l, -2.0, ((n * (U_42_ - U)) * (l / Om)))))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.9e-36)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 5e-309)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(Float64(fma(Float64(l * Float64(n / Om)), Float64(U_42_ - U), Float64(l * -2.0)) / Om) * Float64(n * l))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(U_42_ - U)) * Float64(l / Om))))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.9e-36], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 5e-309], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.89999999999999985e-36

   1. Initial program 43.9%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in t around inf 60.0%

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

      1. pow1/2 60.1%

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

      2. distribute-lft-out 60.1%

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

      3. associate-/l* 60.2%

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

      4. associate-/l* 60.2%

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

      5. *-commutative 60.2%

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

      6. *-commutative 60.2%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in U around 0 60.3%

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

   if -1.89999999999999985e-36 < n < 4.9999999999999995e-309

   1. Initial program 43.1%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in t around inf 59.3%

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

      1. pow1/2 59.7%

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

      2. distribute-lft-out 59.7%

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

      3. associate-/l* 58.3%

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

      4. associate-/l* 56.9%

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

      5. *-commutative 56.9%

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

      6. *-commutative 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. *-un-lft-identity 56.9%

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

      2. unpow1/2 56.5%

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

      3. fma-def 56.5%

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

      4. +-commutative 56.5%

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

      5. associate-/r/ 57.8%

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

      6. *-commutative 57.8%

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

      7. associate-*r* 56.4%

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

   8. Applied egg-rr 56.4%

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

      1. *-lft-identity 56.4%

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

      2. fma-udef 56.4%

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

      3. associate-*r* 60.5%

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

      4. associate-/r/ 56.4%

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

      5. associate-*r* 57.2%

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

      6. distribute-rgt-out 57.1%

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

   10. Simplified 70.0%

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

   if 4.9999999999999995e-309 < n

   1. Initial program 45.9%

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

   3. Simplified 56.6%

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

      1. sqrt-prod 69.7%

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

   5. Applied egg-rr 69.7%

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

      1. *-commutative 69.7%

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

      2. *-commutative 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(n \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array} \]

Alternative 4: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-34}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{U \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1e-34)
   (pow
    (*
     2.0
     (+
      (* n (* t U))
      (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
    0.5)
   (if (<= n 1.85e+111)
     (sqrt
      (*
       2.0
       (*
        U
        (+
         (* n t)
         (* (/ (fma (* l (/ n Om)) (- U* U) (* l -2.0)) Om) (* n l))))))
     (if (<= n 3.6e+197)
       (* (sqrt (* n 2.0)) (sqrt (* t U)))
       (*
        (* l (sqrt 2.0))
        (* n (sqrt (/ (* U (- (/ U* Om) (/ U Om))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1e-34) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 1.85e+111) {
		tmp = sqrt((2.0 * (U * ((n * t) + ((fma((l * (n / Om)), (U_42_ - U), (l * -2.0)) / Om) * (n * l))))));
	} else if (n <= 3.6e+197) {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	} else {
		tmp = (l * sqrt(2.0)) * (n * sqrt(((U * ((U_42_ / Om) - (U / Om))) / Om)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1e-34)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 1.85e+111)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(Float64(fma(Float64(l * Float64(n / Om)), Float64(U_42_ - U), Float64(l * -2.0)) / Om) * Float64(n * l))))));
	elseif (n <= 3.6e+197)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(n * sqrt(Float64(Float64(U * Float64(Float64(U_42_ / Om) - Float64(U / Om))) / Om))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1e-34], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 1.85e+111], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.6e+197], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[N[(N[(U * N[(N[(U$42$ / Om), $MachinePrecision] - N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.99999999999999928e-35

   1. Initial program 43.9%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in t around inf 60.0%

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

      1. pow1/2 60.1%

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

      2. distribute-lft-out 60.1%

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

      3. associate-/l* 60.2%

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

      4. associate-/l* 60.2%

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

      5. *-commutative 60.2%

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

      6. *-commutative 60.2%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in U around 0 60.3%

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

   if -9.99999999999999928e-35 < n < 1.8500000000000001e111

   1. Initial program 43.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in t around inf 55.9%

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

      1. pow1/2 56.3%

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

      2. distribute-lft-out 56.3%

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

      3. associate-/l* 54.5%

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

      4. associate-/l* 54.4%

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

      5. *-commutative 54.4%

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

      6. *-commutative 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. *-un-lft-identity 54.4%

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

      2. unpow1/2 54.0%

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

      3. fma-def 54.0%

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

      4. +-commutative 54.0%

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

      5. associate-/r/ 54.7%

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

      6. *-commutative 54.7%

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

      7. associate-*r* 54.1%

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

   8. Applied egg-rr 54.1%

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

      1. *-lft-identity 54.1%

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

      2. fma-udef 54.1%

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

      3. associate-*r* 56.4%

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

      4. associate-/r/ 55.9%

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

      5. associate-*r* 56.9%

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

      6. distribute-rgt-out 56.8%

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

   10. Simplified 65.5%

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

   if 1.8500000000000001e111 < n < 3.59999999999999982e197

   1. Initial program 49.1%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around inf 42.3%

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

      1. sqrt-prod 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 3.59999999999999982e197 < n

   1. Initial program 52.6%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in t around inf 52.6%

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

   5. Taylor expanded in l around inf 37.6%

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

   6. Taylor expanded in n around inf 41.9%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-34}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \frac{\mathsf{fma}\left(\ell \cdot \frac{n}{Om}, U* - U, \ell \cdot -2\right)}{Om} \cdot \left(n \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{U \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)}{Om}}\right)\\ \end{array} \]

Alternative 5: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{\frac{Om}{\ell \cdot U*}}, \frac{\ell}{Om}, t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -8.2e-80)
   (pow
    (*
     2.0
     (+
      (* n (* t U))
      (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
    0.5)
   (if (<= n 4.4e+110)
     (sqrt
      (*
       2.0
       (*
        U
        (- (* n t) (* (* n l) (/ (* l (- 2.0 (/ (* n (- U* U)) Om))) Om))))))
     (if (<= n 3.6e+197)
       (* (sqrt (* n 2.0)) (sqrt (* t U)))
       (sqrt (* 2.0 (* (* n U) (fma (/ n (/ Om (* l U*))) (/ l Om) t))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -8.2e-80) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 4.4e+110) {
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 3.6e+197) {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	} else {
		tmp = sqrt((2.0 * ((n * U) * fma((n / (Om / (l * U_42_))), (l / Om), t))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -8.2e-80)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 4.4e+110)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) - Float64(Float64(n * l) * Float64(Float64(l * Float64(2.0 - Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 3.6e+197)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(Float64(n / Float64(Om / Float64(l * U_42_))), Float64(l / Om), t))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -8.2e-80], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 4.4e+110], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] - N[(N[(n * l), $MachinePrecision] * N[(N[(l * N[(2.0 - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.6e+197], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.1999999999999999e-80

   1. Initial program 44.9%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in t around inf 60.4%

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

      1. pow1/2 60.5%

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

      2. distribute-lft-out 60.5%

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

      3. associate-/l* 60.6%

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

      4. associate-/l* 60.6%

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

      5. *-commutative 60.6%

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

      6. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U around 0 60.7%

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

   if -8.1999999999999999e-80 < n < 4.39999999999999984e110

   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 58.6%

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

   4. Taylor expanded in t around inf 55.3%

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

      1. pow1/2 55.7%

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

      2. distribute-lft-out 55.7%

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

      3. associate-/l* 53.7%

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

      4. associate-/l* 53.7%

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

      5. *-commutative 53.7%

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

      6. *-commutative 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. unpow1/2 53.2%

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

      3. fma-def 53.2%

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

      4. +-commutative 53.2%

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

      5. associate-/r/ 54.0%

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

      6. *-commutative 54.0%

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

      7. associate-*r* 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-lft-identity 53.3%

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

      2. fma-udef 53.3%

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

      3. associate-*r* 55.9%

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

      4. associate-/r/ 55.3%

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

      5. associate-*r* 56.4%

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

      6. distribute-rgt-out 56.4%

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

   10. Simplified 65.3%

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

   11. Taylor expanded in l around 0 64.6%

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

   if 4.39999999999999984e110 < n < 3.59999999999999982e197

   1. Initial program 49.1%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around inf 42.3%

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

      1. sqrt-prod 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 3.59999999999999982e197 < n

   1. Initial program 52.6%

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

   3. Simplified 56.5%

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

      1. *-un-lft-identity 56.5%

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

      2. associate-*l* 56.5%

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

   5. Applied egg-rr 56.5%

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

      1. *-lft-identity 56.5%

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

      2. associate-*r* 60.8%

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

      3. +-commutative 60.8%

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

      4. *-commutative 60.8%

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

      5. fma-udef 60.8%

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

      6. *-commutative 60.8%

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

      7. associate-*l/ 60.9%

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

      8. associate-*r* 61.4%

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

      9. *-commutative 61.4%

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

      10. associate-*r* 61.5%

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

      11. +-commutative 61.5%

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

      12. fma-def 61.5%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in U* around inf 61.8%

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

      1. associate-/l* 61.8%

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

   10. Simplified 61.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{n}{\frac{Om}{\ell \cdot U*}}, \frac{\ell}{Om}, t\right)\right)}\\ \end{array} \]

Alternative 6: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{U \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -8.8e-80)
   (pow
    (*
     2.0
     (+
      (* n (* t U))
      (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
    0.5)
   (if (<= n 1.2e+111)
     (sqrt
      (*
       2.0
       (*
        U
        (- (* n t) (* (* n l) (/ (* l (- 2.0 (/ (* n (- U* U)) Om))) Om))))))
     (if (<= n 5.1e+197)
       (* (sqrt (* n 2.0)) (sqrt (* t U)))
       (*
        (* l (sqrt 2.0))
        (* n (sqrt (/ (* U (- (/ U* Om) (/ U Om))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -8.8e-80) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 1.2e+111) {
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 5.1e+197) {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	} else {
		tmp = (l * sqrt(2.0)) * (n * sqrt(((U * ((U_42_ / Om) - (U / Om))) / Om)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-8.8d-80)) then
        tmp = (2.0d0 * ((n * (t * u)) + ((n * (l * (u * (((n * (l * u_42)) / om) + (l * (-2.0d0)))))) / om))) ** 0.5d0
    else if (n <= 1.2d+111) then
        tmp = sqrt((2.0d0 * (u * ((n * t) - ((n * l) * ((l * (2.0d0 - ((n * (u_42 - u)) / om))) / om))))))
    else if (n <= 5.1d+197) then
        tmp = sqrt((n * 2.0d0)) * sqrt((t * u))
    else
        tmp = (l * sqrt(2.0d0)) * (n * sqrt(((u * ((u_42 / om) - (u / om))) / om)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -8.8e-80) {
		tmp = Math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 1.2e+111) {
		tmp = Math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 5.1e+197) {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((t * U));
	} else {
		tmp = (l * Math.sqrt(2.0)) * (n * Math.sqrt(((U * ((U_42_ / Om) - (U / Om))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -8.8e-80:
		tmp = math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5)
	elif n <= 1.2e+111:
		tmp = math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))))
	elif n <= 5.1e+197:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((t * U))
	else:
		tmp = (l * math.sqrt(2.0)) * (n * math.sqrt(((U * ((U_42_ / Om) - (U / Om))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -8.8e-80)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 1.2e+111)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) - Float64(Float64(n * l) * Float64(Float64(l * Float64(2.0 - Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 5.1e+197)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(n * sqrt(Float64(Float64(U * Float64(Float64(U_42_ / Om) - Float64(U / Om))) / Om))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -8.8e-80)
		tmp = (2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 1.2e+111)
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 5.1e+197)
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	else
		tmp = (l * sqrt(2.0)) * (n * sqrt(((U * ((U_42_ / Om) - (U / Om))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -8.8e-80], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 1.2e+111], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] - N[(N[(n * l), $MachinePrecision] * N[(N[(l * N[(2.0 - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 5.1e+197], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[N[(N[(U * N[(N[(U$42$ / Om), $MachinePrecision] - N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.80000000000000041e-80

   1. Initial program 44.9%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in t around inf 60.4%

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

      1. pow1/2 60.5%

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

      2. distribute-lft-out 60.5%

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

      3. associate-/l* 60.6%

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

      4. associate-/l* 60.6%

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

      5. *-commutative 60.6%

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

      6. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U around 0 60.7%

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

   if -8.80000000000000041e-80 < n < 1.20000000000000003e111

   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 58.6%

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

   4. Taylor expanded in t around inf 55.3%

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

      1. pow1/2 55.7%

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

      2. distribute-lft-out 55.7%

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

      3. associate-/l* 53.7%

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

      4. associate-/l* 53.7%

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

      5. *-commutative 53.7%

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

      6. *-commutative 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. unpow1/2 53.2%

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

      3. fma-def 53.2%

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

      4. +-commutative 53.2%

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

      5. associate-/r/ 54.0%

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

      6. *-commutative 54.0%

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

      7. associate-*r* 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-lft-identity 53.3%

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

      2. fma-udef 53.3%

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

      3. associate-*r* 55.9%

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

      4. associate-/r/ 55.3%

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

      5. associate-*r* 56.4%

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

      6. distribute-rgt-out 56.4%

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

   10. Simplified 65.3%

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

   11. Taylor expanded in l around 0 64.6%

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

   if 1.20000000000000003e111 < n < 5.09999999999999992e197

   1. Initial program 49.1%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around inf 42.3%

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

      1. sqrt-prod 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 5.09999999999999992e197 < n

   1. Initial program 52.6%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in t around inf 52.6%

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

   5. Taylor expanded in l around inf 37.6%

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

   6. Taylor expanded in n around inf 41.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{U \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)}{Om}}\right)\\ \end{array} \]

Alternative 7: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{\frac{U \cdot U*}{Om}}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -3.1e-80)
   (pow
    (*
     2.0
     (+
      (* n (* t U))
      (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
    0.5)
   (if (<= n 1.2e+111)
     (sqrt
      (*
       2.0
       (*
        U
        (- (* n t) (* (* n l) (/ (* l (- 2.0 (/ (* n (- U* U)) Om))) Om))))))
     (if (<= n 3.6e+197)
       (* (sqrt (* n 2.0)) (sqrt (* t U)))
       (* (* l (sqrt 2.0)) (* n (sqrt (/ (/ (* U U*) Om) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -3.1e-80) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 1.2e+111) {
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 3.6e+197) {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	} else {
		tmp = (l * sqrt(2.0)) * (n * sqrt((((U * U_42_) / Om) / Om)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-3.1d-80)) then
        tmp = (2.0d0 * ((n * (t * u)) + ((n * (l * (u * (((n * (l * u_42)) / om) + (l * (-2.0d0)))))) / om))) ** 0.5d0
    else if (n <= 1.2d+111) then
        tmp = sqrt((2.0d0 * (u * ((n * t) - ((n * l) * ((l * (2.0d0 - ((n * (u_42 - u)) / om))) / om))))))
    else if (n <= 3.6d+197) then
        tmp = sqrt((n * 2.0d0)) * sqrt((t * u))
    else
        tmp = (l * sqrt(2.0d0)) * (n * sqrt((((u * u_42) / om) / om)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -3.1e-80) {
		tmp = Math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else if (n <= 1.2e+111) {
		tmp = Math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 3.6e+197) {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((t * U));
	} else {
		tmp = (l * Math.sqrt(2.0)) * (n * Math.sqrt((((U * U_42_) / Om) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -3.1e-80:
		tmp = math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5)
	elif n <= 1.2e+111:
		tmp = math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))))
	elif n <= 3.6e+197:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((t * U))
	else:
		tmp = (l * math.sqrt(2.0)) * (n * math.sqrt((((U * U_42_) / Om) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -3.1e-80)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 1.2e+111)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) - Float64(Float64(n * l) * Float64(Float64(l * Float64(2.0 - Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 3.6e+197)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(n * sqrt(Float64(Float64(Float64(U * U_42_) / Om) / Om))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -3.1e-80)
		tmp = (2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))) ^ 0.5;
	elseif (n <= 1.2e+111)
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 3.6e+197)
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	else
		tmp = (l * sqrt(2.0)) * (n * sqrt((((U * U_42_) / Om) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -3.1e-80], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 1.2e+111], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] - N[(N[(n * l), $MachinePrecision] * N[(N[(l * N[(2.0 - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.6e+197], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[Sqrt[N[(N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.10000000000000016e-80

   1. Initial program 44.9%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in t around inf 60.4%

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

      1. pow1/2 60.5%

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

      2. distribute-lft-out 60.5%

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

      3. associate-/l* 60.6%

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

      4. associate-/l* 60.6%

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

      5. *-commutative 60.6%

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

      6. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U around 0 60.7%

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

   if -3.10000000000000016e-80 < n < 1.20000000000000003e111

   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 58.6%

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

   4. Taylor expanded in t around inf 55.3%

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

      1. pow1/2 55.7%

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

      2. distribute-lft-out 55.7%

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

      3. associate-/l* 53.7%

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

      4. associate-/l* 53.7%

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

      5. *-commutative 53.7%

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

      6. *-commutative 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. unpow1/2 53.2%

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

      3. fma-def 53.2%

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

      4. +-commutative 53.2%

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

      5. associate-/r/ 54.0%

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

      6. *-commutative 54.0%

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

      7. associate-*r* 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-lft-identity 53.3%

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

      2. fma-udef 53.3%

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

      3. associate-*r* 55.9%

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

      4. associate-/r/ 55.3%

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

      5. associate-*r* 56.4%

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

      6. distribute-rgt-out 56.4%

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

   10. Simplified 65.3%

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

   11. Taylor expanded in l around 0 64.6%

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

   if 1.20000000000000003e111 < n < 3.59999999999999982e197

   1. Initial program 49.1%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around inf 42.3%

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

      1. sqrt-prod 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 3.59999999999999982e197 < n

   1. Initial program 52.6%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in t around inf 52.6%

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

   5. Taylor expanded in l around inf 37.6%

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

   6. Taylor expanded in n around inf 41.9%

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

   7. Taylor expanded in U* around inf 42.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \left(n \cdot \sqrt{\frac{\frac{U \cdot U*}{Om}}{Om}}\right)\\ \end{array} \]

Alternative 8: 59.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\\ \mathbf{if}\;n \leq -5 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot t_1\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot t_1}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ (/ (* n (* l U*)) Om) (* l -2.0))))
   (if (<= n -5e-80)
     (pow (* 2.0 (+ (* n (* t U)) (/ (* n (* l (* U t_1))) Om))) 0.5)
     (if (<= n 1.2e+111)
       (sqrt
        (*
         2.0
         (*
          U
          (- (* n t) (* (* n l) (/ (* l (- 2.0 (/ (* n (- U* U)) Om))) Om))))))
       (if (<= n 3.6e+197)
         (* (sqrt (* n 2.0)) (sqrt (* t U)))
         (sqrt (* (* n 2.0) (* U (+ t (/ (* l t_1) Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((n * (l * U_42_)) / Om) + (l * -2.0);
	double tmp;
	if (n <= -5e-80) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * t_1))) / Om))), 0.5);
	} else if (n <= 1.2e+111) {
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 3.6e+197) {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * t_1) / Om)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((n * (l * u_42)) / om) + (l * (-2.0d0))
    if (n <= (-5d-80)) then
        tmp = (2.0d0 * ((n * (t * u)) + ((n * (l * (u * t_1))) / om))) ** 0.5d0
    else if (n <= 1.2d+111) then
        tmp = sqrt((2.0d0 * (u * ((n * t) - ((n * l) * ((l * (2.0d0 - ((n * (u_42 - u)) / om))) / om))))))
    else if (n <= 3.6d+197) then
        tmp = sqrt((n * 2.0d0)) * sqrt((t * u))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * t_1) / om)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((n * (l * U_42_)) / Om) + (l * -2.0);
	double tmp;
	if (n <= -5e-80) {
		tmp = Math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * t_1))) / Om))), 0.5);
	} else if (n <= 1.2e+111) {
		tmp = Math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	} else if (n <= 3.6e+197) {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((t * U));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * t_1) / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = ((n * (l * U_42_)) / Om) + (l * -2.0)
	tmp = 0
	if n <= -5e-80:
		tmp = math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * t_1))) / Om))), 0.5)
	elif n <= 1.2e+111:
		tmp = math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))))
	elif n <= 3.6e+197:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((t * U))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * t_1) / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))
	tmp = 0.0
	if (n <= -5e-80)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * t_1))) / Om))) ^ 0.5;
	elseif (n <= 1.2e+111)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) - Float64(Float64(n * l) * Float64(Float64(l * Float64(2.0 - Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 3.6e+197)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * t_1) / Om)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((n * (l * U_42_)) / Om) + (l * -2.0);
	tmp = 0.0;
	if (n <= -5e-80)
		tmp = (2.0 * ((n * (t * U)) + ((n * (l * (U * t_1))) / Om))) ^ 0.5;
	elseif (n <= 1.2e+111)
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	elseif (n <= 3.6e+197)
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	else
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * t_1) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5e-80], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 1.2e+111], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] - N[(N[(n * l), $MachinePrecision] * N[(N[(l * N[(2.0 - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.6e+197], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5e-80

   1. Initial program 44.9%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in t around inf 60.4%

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

      1. pow1/2 60.5%

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

      2. distribute-lft-out 60.5%

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

      3. associate-/l* 60.6%

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

      4. associate-/l* 60.6%

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

      5. *-commutative 60.6%

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

      6. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U around 0 60.7%

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

   if -5e-80 < n < 1.20000000000000003e111

   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 58.6%

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

   4. Taylor expanded in t around inf 55.3%

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

      1. pow1/2 55.7%

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

      2. distribute-lft-out 55.7%

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

      3. associate-/l* 53.7%

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

      4. associate-/l* 53.7%

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

      5. *-commutative 53.7%

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

      6. *-commutative 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. unpow1/2 53.2%

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

      3. fma-def 53.2%

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

      4. +-commutative 53.2%

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

      5. associate-/r/ 54.0%

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

      6. *-commutative 54.0%

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

      7. associate-*r* 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-lft-identity 53.3%

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

      2. fma-udef 53.3%

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

      3. associate-*r* 55.9%

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

      4. associate-/r/ 55.3%

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

      5. associate-*r* 56.4%

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

      6. distribute-rgt-out 56.4%

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

   10. Simplified 65.3%

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

   11. Taylor expanded in l around 0 64.6%

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

   if 1.20000000000000003e111 < n < 3.59999999999999982e197

   1. Initial program 49.1%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in t around inf 42.3%

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

      1. sqrt-prod 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 3.59999999999999982e197 < n

   1. Initial program 52.6%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in U around 0 57.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-80}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 9: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e+67)
   (pow
    (*
     2.0
     (+
      (* n (* t U))
      (/ (* n (* l (* U (+ (/ (* n (* l U*)) Om) (* l -2.0))))) Om)))
    0.5)
   (sqrt
    (*
     2.0
     (*
      U
      (- (* n t) (* (* n l) (/ (* l (- 2.0 (/ (* n (- U* U)) Om))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+67) {
		tmp = pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.15d+67) then
        tmp = (2.0d0 * ((n * (t * u)) + ((n * (l * (u * (((n * (l * u_42)) / om) + (l * (-2.0d0)))))) / om))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * ((n * t) - ((n * l) * ((l * (2.0d0 - ((n * (u_42 - u)) / om))) / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+67) {
		tmp = Math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e+67:
		tmp = math.pow((2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e+67)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(t * U)) + Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))))) / Om))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) - Float64(Float64(n * l) * Float64(Float64(l * Float64(2.0 - Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e+67)
		tmp = (2.0 * ((n * (t * U)) + ((n * (l * (U * (((n * (l * U_42_)) / Om) + (l * -2.0))))) / Om))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e+67], N[Power[N[(2.0 * N[(N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(l * N[(U * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] - N[(N[(n * l), $MachinePrecision] * N[(N[(l * N[(2.0 - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.1499999999999999e67

   1. Initial program 48.5%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in t around inf 54.8%

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

      1. pow1/2 54.9%

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

      2. distribute-lft-out 54.9%

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

      3. associate-/l* 53.6%

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

      4. associate-/l* 54.0%

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

      5. *-commutative 54.0%

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

      6. *-commutative 54.0%

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

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in U around 0 54.6%

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

   if 1.1499999999999999e67 < l

   1. Initial program 26.3%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in t around inf 55.6%

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

      1. pow1/2 56.6%

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

      2. distribute-lft-out 56.6%

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

      3. associate-/l* 56.6%

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

      4. associate-/l* 58.8%

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

      5. *-commutative 58.8%

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

      6. *-commutative 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. unpow1/2 58.0%

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

      3. fma-def 58.0%

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

      4. +-commutative 58.0%

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

      5. associate-/r/ 58.0%

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

      6. *-commutative 58.0%

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

      7. associate-*r* 58.0%

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

   8. Applied egg-rr 58.0%

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

      1. *-lft-identity 58.0%

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

      2. fma-udef 58.0%

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

      3. associate-*r* 53.5%

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

      4. associate-/r/ 51.2%

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

      5. associate-*r* 51.2%

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

      6. distribute-rgt-out 51.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in l around 0 65.2%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)\right)\right)}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 10: 58.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.3 \cdot 10^{+108} \lor \neg \left(Om \leq 1.25 \cdot 10^{-53}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \frac{\ell \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1.3e+108) (not (<= Om 1.25e-53)))
   (sqrt (* 2.0 (* U (+ (* n t) (* (* n l) (/ (* l -2.0) Om))))))
   (sqrt
    (*
     (* n 2.0)
     (* U (+ t (/ (* l (+ (/ (* n (* l U*)) Om) (* l -2.0))) Om)))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.3e+108) || !(Om <= 1.25e-53)) {
		tmp = sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.3d+108)) .or. (.not. (om <= 1.25d-53))) then
        tmp = sqrt((2.0d0 * (u * ((n * t) + ((n * l) * ((l * (-2.0d0)) / om))))))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * (((n * (l * u_42)) / om) + (l * (-2.0d0)))) / om)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.3e+108) || !(Om <= 1.25e-53)) {
		tmp = Math.sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1.3e+108) or not (Om <= 1.25e-53):
		tmp = math.sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.3e+108) || !(Om <= 1.25e-53))
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(Float64(n * l) * Float64(Float64(l * -2.0) / Om))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))) / Om)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.3e+108) || ~((Om <= 1.25e-53)))
		tmp = sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))));
	else
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.3e+108], N[Not[LessEqual[Om, 1.25e-53]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(n * l), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.3000000000000001e108 or 1.25e-53 < Om

   1. Initial program 44.6%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in t around inf 50.9%

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

      1. pow1/2 51.4%

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

      2. distribute-lft-out 51.4%

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

      3. associate-/l* 49.3%

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

      4. associate-/l* 50.7%

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

      5. *-commutative 50.7%

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

      6. *-commutative 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. *-un-lft-identity 50.7%

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

      2. unpow1/2 50.3%

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

      3. fma-def 50.3%

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

      4. +-commutative 50.3%

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

      5. associate-/r/ 51.0%

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

      6. *-commutative 51.0%

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

      7. associate-*r* 51.0%

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

   8. Applied egg-rr 51.0%

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

      1. *-lft-identity 51.0%

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

      2. fma-udef 51.0%

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

      3. associate-*r* 50.4%

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

      4. associate-/r/ 53.9%

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

      5. associate-*r* 53.9%

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

      6. distribute-rgt-out 54.0%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in n around 0 59.3%

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

       1. *-commutative 59.3%

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

   13. Simplified 59.3%

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

   if -1.3000000000000001e108 < Om < 1.25e-53

   1. Initial program 44.9%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in U around 0 59.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.3 \cdot 10^{+108} \lor \neg \left(Om \leq 1.25 \cdot 10^{-53}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \frac{\ell \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 11: 60.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.3e+67)
   (sqrt
    (*
     (* n 2.0)
     (* U (+ t (/ (* l (+ (/ (* n (* l U*)) Om) (* l -2.0))) Om)))))
   (sqrt
    (*
     2.0
     (*
      U
      (- (* n t) (* (* n l) (/ (* l (- 2.0 (/ (* n (- U* U)) Om))) Om))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.3e+67) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	} else {
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.3d+67) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * (((n * (l * u_42)) / om) + (l * (-2.0d0)))) / om)))))
    else
        tmp = sqrt((2.0d0 * (u * ((n * t) - ((n * l) * ((l * (2.0d0 - ((n * (u_42 - u)) / om))) / om))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.3e+67) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.3e+67:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))))
	else:
		tmp = math.sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.3e+67)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(Float64(n * Float64(l * U_42_)) / Om) + Float64(l * -2.0))) / Om)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) - Float64(Float64(n * l) * Float64(Float64(l * Float64(2.0 - Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.3e+67)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * (((n * (l * U_42_)) / Om) + (l * -2.0))) / Om)))));
	else
		tmp = sqrt((2.0 * (U * ((n * t) - ((n * l) * ((l * (2.0 - ((n * (U_42_ - U)) / Om))) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.3e+67], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] - N[(N[(n * l), $MachinePrecision] * N[(N[(l * N[(2.0 - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2999999999999999e67

   1. Initial program 48.5%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in U around 0 52.6%

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

   if 2.2999999999999999e67 < l

   1. Initial program 26.3%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in t around inf 55.6%

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

      1. pow1/2 56.6%

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

      2. distribute-lft-out 56.6%

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

      3. associate-/l* 56.6%

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

      4. associate-/l* 58.8%

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

      5. *-commutative 58.8%

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

      6. *-commutative 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. unpow1/2 58.0%

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

      3. fma-def 58.0%

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

      4. +-commutative 58.0%

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

      5. associate-/r/ 58.0%

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

      6. *-commutative 58.0%

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

      7. associate-*r* 58.0%

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

   8. Applied egg-rr 58.0%

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

      1. *-lft-identity 58.0%

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

      2. fma-udef 58.0%

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

      3. associate-*r* 53.5%

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

      4. associate-/r/ 51.2%

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

      5. associate-*r* 51.2%

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

      6. distribute-rgt-out 51.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in l around 0 65.2%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\frac{n \cdot \left(\ell \cdot U*\right)}{Om} + \ell \cdot -2\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t - \left(n \cdot \ell\right) \cdot \frac{\ell \cdot \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]

Alternative 12: 39.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{+52} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+79}\right) \land \ell \leq 6 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l 3.2e+52) (and (not (<= l 1.7e+79)) (<= l 6e+111)))
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* 2.0 (* -2.0 (/ (* U (* n (* l l))) Om))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= 3.2e+52) || (!(l <= 1.7e+79) && (l <= 6e+111))) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((2.0 * (-2.0 * ((U * (n * (l * l))) / Om))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((l <= 3.2d+52) .or. (.not. (l <= 1.7d+79)) .and. (l <= 6d+111)) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((2.0d0 * ((-2.0d0) * ((u * (n * (l * l))) / om))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= 3.2e+52) || (!(l <= 1.7e+79) && (l <= 6e+111))) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((2.0 * (-2.0 * ((U * (n * (l * l))) / Om))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= 3.2e+52) or (not (l <= 1.7e+79) and (l <= 6e+111)):
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((2.0 * (-2.0 * ((U * (n * (l * l))) / Om))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= 3.2e+52) || (!(l <= 1.7e+79) && (l <= 6e+111)))
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(-2.0 * Float64(Float64(U * Float64(n * Float64(l * l))) / Om))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= 3.2e+52) || (~((l <= 1.7e+79)) && (l <= 6e+111)))
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((2.0 * (-2.0 * ((U * (n * (l * l))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, 3.2e+52], And[N[Not[LessEqual[l, 1.7e+79]], $MachinePrecision], LessEqual[l, 6e+111]]], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(-2.0 * N[(N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.2e52 or 1.70000000000000016e79 < l < 6e111

   1. Initial program 46.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around inf 35.7%

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

   5. Taylor expanded in n around 0 35.7%

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

      1. associate-*r* 37.5%

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

   7. Simplified 37.5%

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

   if 3.2e52 < l < 1.70000000000000016e79 or 6e111 < l

   1. Initial program 32.9%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in l around inf 48.8%

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

      1. associate-*r* 48.7%

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

      2. unpow2 48.7%

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

      3. sub-neg 48.7%

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

      4. unpow2 48.7%

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

      5. associate-*r/ 48.7%

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

      6. metadata-eval 48.7%

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

      7. distribute-neg-frac 48.7%

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

      8. metadata-eval 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in U around 0 48.8%

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

      1. associate-*r* 48.7%

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

      2. unpow2 48.7%

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

      3. *-commutative 48.7%

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

      4. unpow2 48.7%

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

      5. associate-*r/ 48.7%

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

      6. metadata-eval 48.7%

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

   9. Simplified 48.7%

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

   10. Taylor expanded in n around 0 29.9%

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

       1. associate-*r* 32.2%

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

       2. unpow2 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{+52} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+79}\right) \land \ell \leq 6 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 13: 39.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := n \cdot \left(\ell \cdot \ell\right)\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{if}\;\ell \leq 1.08 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(U \cdot \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+111}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U \cdot t_1}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* l l))) (t_2 (sqrt (* 2.0 (* U (* n t))))))
   (if (<= l 1.08e+52)
     t_2
     (if (<= l 9.8e+78)
       (sqrt (* 2.0 (* t_1 (* U (/ -2.0 Om)))))
       (if (<= l 4.2e+111) t_2 (sqrt (* 2.0 (* -2.0 (/ (* U t_1) Om)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (l * l);
	double t_2 = sqrt((2.0 * (U * (n * t))));
	double tmp;
	if (l <= 1.08e+52) {
		tmp = t_2;
	} else if (l <= 9.8e+78) {
		tmp = sqrt((2.0 * (t_1 * (U * (-2.0 / Om)))));
	} else if (l <= 4.2e+111) {
		tmp = t_2;
	} else {
		tmp = sqrt((2.0 * (-2.0 * ((U * t_1) / Om))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = n * (l * l)
    t_2 = sqrt((2.0d0 * (u * (n * t))))
    if (l <= 1.08d+52) then
        tmp = t_2
    else if (l <= 9.8d+78) then
        tmp = sqrt((2.0d0 * (t_1 * (u * ((-2.0d0) / om)))))
    else if (l <= 4.2d+111) then
        tmp = t_2
    else
        tmp = sqrt((2.0d0 * ((-2.0d0) * ((u * t_1) / om))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (l * l);
	double t_2 = Math.sqrt((2.0 * (U * (n * t))));
	double tmp;
	if (l <= 1.08e+52) {
		tmp = t_2;
	} else if (l <= 9.8e+78) {
		tmp = Math.sqrt((2.0 * (t_1 * (U * (-2.0 / Om)))));
	} else if (l <= 4.2e+111) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt((2.0 * (-2.0 * ((U * t_1) / Om))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = n * (l * l)
	t_2 = math.sqrt((2.0 * (U * (n * t))))
	tmp = 0
	if l <= 1.08e+52:
		tmp = t_2
	elif l <= 9.8e+78:
		tmp = math.sqrt((2.0 * (t_1 * (U * (-2.0 / Om)))))
	elif l <= 4.2e+111:
		tmp = t_2
	else:
		tmp = math.sqrt((2.0 * (-2.0 * ((U * t_1) / Om))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(l * l))
	t_2 = sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
	tmp = 0.0
	if (l <= 1.08e+52)
		tmp = t_2;
	elseif (l <= 9.8e+78)
		tmp = sqrt(Float64(2.0 * Float64(t_1 * Float64(U * Float64(-2.0 / Om)))));
	elseif (l <= 4.2e+111)
		tmp = t_2;
	else
		tmp = sqrt(Float64(2.0 * Float64(-2.0 * Float64(Float64(U * t_1) / Om))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (l * l);
	t_2 = sqrt((2.0 * (U * (n * t))));
	tmp = 0.0;
	if (l <= 1.08e+52)
		tmp = t_2;
	elseif (l <= 9.8e+78)
		tmp = sqrt((2.0 * (t_1 * (U * (-2.0 / Om)))));
	elseif (l <= 4.2e+111)
		tmp = t_2;
	else
		tmp = sqrt((2.0 * (-2.0 * ((U * t_1) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1.08e+52], t$95$2, If[LessEqual[l, 9.8e+78], N[Sqrt[N[(2.0 * N[(t$95$1 * N[(U * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.2e+111], t$95$2, N[Sqrt[N[(2.0 * N[(-2.0 * N[(N[(U * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.07999999999999997e52 or 9.8000000000000004e78 < l < 4.1999999999999999e111

   1. Initial program 46.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around inf 35.7%

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

   5. Taylor expanded in n around 0 35.7%

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

      1. associate-*r* 37.5%

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

   7. Simplified 37.5%

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

   if 1.07999999999999997e52 < l < 9.8000000000000004e78

   1. Initial program 68.0%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in l around inf 83.3%

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

      1. associate-*r* 83.3%

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

      2. unpow2 83.3%

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

      3. sub-neg 83.3%

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

      4. unpow2 83.3%

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

      5. associate-*r/ 83.3%

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

      6. metadata-eval 83.3%

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

      7. distribute-neg-frac 83.3%

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

      8. metadata-eval 83.3%

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

   6. Simplified 83.3%

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

   7. Taylor expanded in n around 0 66.7%

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

      1. associate-*r/ 66.7%

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

      2. associate-*l/ 66.7%

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

      3. *-commutative 66.7%

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

   9. Simplified 66.7%

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

   if 4.1999999999999999e111 < l

   1. Initial program 26.9%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in l around inf 42.9%

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

      1. associate-*r* 42.8%

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

      2. unpow2 42.8%

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

      3. sub-neg 42.8%

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

      4. unpow2 42.8%

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

      5. associate-*r/ 42.8%

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

      6. metadata-eval 42.8%

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

      7. distribute-neg-frac 42.8%

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

      8. metadata-eval 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in U around 0 42.9%

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

      1. associate-*r* 42.8%

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

      2. unpow2 42.8%

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

      3. *-commutative 42.8%

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

      4. unpow2 42.8%

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

      5. associate-*r/ 42.8%

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

      6. metadata-eval 42.8%

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

   9. Simplified 42.8%

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

   10. Taylor expanded in n around 0 23.6%

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

       1. associate-*r* 26.3%

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

       2. unpow2 26.3%

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

   12. Simplified 26.3%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.08 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \frac{-2}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]

Alternative 14: 49.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U* \leq 4 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \frac{\ell \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 4e+162)
   (sqrt (* 2.0 (* U (+ (* n t) (* (* n l) (/ (* l -2.0) Om))))))
   (sqrt (* (* n 2.0) (* 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_) {
	double tmp;
	if (U_42_ <= 4e+162) {
		tmp = sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t + (-2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 4d+162) then
        tmp = sqrt((2.0d0 * (u * ((n * t) + ((n * l) * ((l * (-2.0d0)) / om))))))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 4e+162) {
		tmp = Math.sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + (-2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 4e+162:
		tmp = math.sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t + (-2.0 * (l * (l / Om)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 4e+162)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(n * t) + Float64(Float64(n * l) * Float64(Float64(l * -2.0) / Om))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 4e+162)
		tmp = sqrt((2.0 * (U * ((n * t) + ((n * l) * ((l * -2.0) / Om))))));
	else
		tmp = sqrt(((n * 2.0) * (U * (t + (-2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 4e+162], N[Sqrt[N[(2.0 * N[(U * N[(N[(n * t), $MachinePrecision] + N[(N[(n * l), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 3.9999999999999998e162

   1. Initial program 43.8%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in t around inf 55.6%

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

      1. pow1/2 55.9%

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

      2. distribute-lft-out 55.9%

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

      3. associate-/l* 54.6%

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

      4. associate-/l* 54.5%

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

      5. *-commutative 54.5%

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

      6. *-commutative 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. *-un-lft-identity 54.5%

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

      2. unpow1/2 54.3%

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

      3. fma-def 54.3%

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

      4. +-commutative 54.3%

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

      5. associate-/r/ 54.7%

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

      6. *-commutative 54.7%

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

      7. associate-*r* 54.3%

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

   8. Applied egg-rr 54.3%

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

      1. *-lft-identity 54.3%

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

      2. fma-udef 54.3%

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

      3. associate-*r* 55.4%

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

      4. associate-/r/ 55.0%

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

      5. associate-*r* 54.9%

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

      6. distribute-rgt-out 54.9%

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

   10. Simplified 59.5%

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

   11. Taylor expanded in n around 0 53.8%

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

       1. *-commutative 53.8%

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

   13. Simplified 53.8%

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

   if 3.9999999999999998e162 < U*

   1. Initial program 50.3%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in n around 0 47.8%

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

      1. *-commutative 47.8%

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

      2. unpow2 47.8%

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

      3. associate-*r/ 53.4%

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

   6. Simplified 53.4%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 4 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t + \left(n \cdot \ell\right) \cdot \frac{\ell \cdot -2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 15: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.6e+102)
   (sqrt (* 2.0 (* (* n U) (+ t (* -2.0 (/ (* l l) Om))))))
   (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 (t <= 1.6e+102) {
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} 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 (t <= 1.6d+102) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((-2.0d0) * ((l * l) / om))))))
    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 (t <= 1.6e+102) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	} 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 t <= 1.6e+102:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))))
	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 (t <= 1.6e+102)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(-2.0 * Float64(Float64(l * l) / Om))))));
	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 (t <= 1.6e+102)
		tmp = sqrt((2.0 * ((n * U) * (t + (-2.0 * ((l * l) / Om))))));
	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[t, 1.6e+102], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $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 t < 1.6e102

   1. Initial program 45.9%

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

   3. Simplified 53.9%

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

      1. *-un-lft-identity 53.9%

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

      2. associate-*l* 53.9%

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

   5. Applied egg-rr 53.9%

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

      1. *-lft-identity 53.9%

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

      2. associate-*r* 56.2%

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

      3. +-commutative 56.2%

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

      4. *-commutative 56.2%

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

      5. fma-udef 56.2%

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

      6. *-commutative 56.2%

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

      7. associate-*l/ 54.2%

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

      8. associate-*r* 55.3%

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

      9. *-commutative 55.3%

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

      10. associate-*r* 52.6%

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

      11. +-commutative 52.6%

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

      12. fma-def 52.6%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in n around 0 43.9%

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

      1. *-commutative 43.9%

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

      2. unpow2 43.9%

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

   10. Simplified 43.9%

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

   if 1.6e102 < t

   1. Initial program 38.7%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in t around inf 48.1%

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

   5. Taylor expanded in n around 0 48.1%

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

      1. associate-*r* 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 16: 47.4% accurate, 2.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

l = abs(l)
function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((n * 2.0) * (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[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 44.7%

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

3. Simplified 55.2%

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

4. Taylor expanded in n around 0 44.5%

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

   1. *-commutative 44.5%

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

   2. unpow2 44.5%

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

   3. associate-*r/ 49.1%

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

6. Simplified 49.1%

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

7. Final simplification 49.1%

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

Alternative 17: 35.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.7%

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

3. Simplified 55.2%

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

   1. *-un-lft-identity 55.2%

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

   2. associate-*l* 55.2%

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

5. Applied egg-rr 55.2%

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

   1. *-lft-identity 55.2%

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

   2. associate-*r* 53.7%

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

   3. +-commutative 53.7%

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

   4. *-commutative 53.7%

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

   5. fma-udef 53.7%

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

   6. *-commutative 53.7%

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

   7. associate-*l/ 52.1%

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

   8. associate-*r* 53.3%

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

   9. *-commutative 53.3%

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

   10. associate-*r* 50.7%

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

   11. +-commutative 50.7%

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

   12. fma-def 50.7%

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

7. Simplified 53.7%

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

8. Taylor expanded in l around 0 30.8%

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

9. Final simplification 30.8%

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

Alternative 18: 35.6% 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 44.7%

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

3. Simplified 55.2%

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

4. Taylor expanded in t around inf 31.5%

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

5. Taylor expanded in n around 0 31.5%

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

   1. associate-*r* 32.6%

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

7. Simplified 32.6%

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

8. Final simplification 32.6%

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

Reproduce

herbie shell --seed 2023213 
(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*))))))