Toniolo and Linder, Equation (13)

Percentage Accurate: 50.9% → 68.5%

Time: 25.4s

Alternatives: 21

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: 50.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 68.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_2 \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\sqrt{t_1 \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_2 5e-318)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
     (if (<= t_2 INFINITY)
       (sqrt
        (* t_1 (+ t (* (/ l Om) (fma l -2.0 (* n (* (/ l Om) (- U* U))))))))
       (*
        (* l (sqrt 2.0))
        (sqrt (* (/ n Om) (* U (+ -2.0 (/ (* n U*) Om))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 5e-318) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U))))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (t + ((l / Om) * fma(l, -2.0, (n * ((l / Om) * (U_42_ - U))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 5e-318)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n / Om) * Float64(U * Float64(-2.0 + Float64(Float64(n * U_42_) / 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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-318], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(n * N[(N[(l / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $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*)))) < 4.9999987e-318

   1. Initial program 13.5%

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

   3. Simplified 38.7%

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

      1. sqrt-prod 42.2%

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

   5. Applied egg-rr 42.2%

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

      1. *-commutative 42.2%

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

      2. *-commutative 42.2%

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

   7. Simplified 42.2%

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

   if 4.9999987e-318 < (*.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 67.9%

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

   3. Simplified 60.5%

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

      1. sqrt-prod 30.9%

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

   5. Applied egg-rr 30.9%

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

      1. *-commutative 30.9%

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

      2. *-commutative 30.9%

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

   7. Simplified 30.9%

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

      1. pow1 30.9%

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

      2. sqrt-unprod 60.5%

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

      3. *-commutative 60.5%

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

      4. associate-*l* 68.3%

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

   9. Applied egg-rr 68.3%

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

       1. unpow1 68.3%

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

       2. associate-*r* 74.5%

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

   11. Simplified 74.5%

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

   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 31.5%

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

   4. Taylor expanded in U around 0 28.9%

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

   5. Taylor expanded in l around 0 29.4%

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

      1. associate-/l* 29.4%

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

      2. unpow2 29.4%

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

      3. sub-neg 29.4%

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

      4. associate-/l* 29.4%

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

      5. metadata-eval 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in t around 0 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. associate-/l* 44.4%

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

      3. *-commutative 44.4%

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

      4. sub-neg 44.4%

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

      5. associate-/l* 41.9%

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

      6. metadata-eval 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. *-lft-identity 41.9%

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

       2. associate-/r/ 44.5%

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

       3. +-commutative 44.5%

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

       4. associate-/l* 47.0%

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

   12. Simplified 47.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 2: 67.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 12.3%

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

   3. Simplified 38.2%

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

      1. sqrt-prod 40.5%

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

   5. Applied egg-rr 40.5%

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

      1. *-commutative 40.5%

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

      2. *-commutative 40.5%

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

   7. Simplified 40.5%

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

      1. pow1 40.5%

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

      2. sqrt-unprod 38.2%

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

      3. *-commutative 38.2%

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

      4. associate-*l* 38.1%

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

   9. Applied egg-rr 38.1%

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

       1. unpow1 38.1%

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

       2. associate-*r* 17.9%

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

   11. Simplified 17.9%

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

   12. Taylor expanded in t around inf 38.0%

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

       1. distribute-lft-out 38.0%

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

       2. *-commutative 38.0%

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

       3. associate-/l* 37.9%

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

       4. +-commutative 37.9%

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

       5. *-commutative 37.9%

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

       6. associate-*r* 39.6%

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

   14. Simplified 39.6%

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

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

   1. Initial program 67.8%

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

   3. Simplified 60.4%

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

      1. sqrt-prod 31.2%

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

   5. Applied egg-rr 31.2%

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

      1. *-commutative 31.2%

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

      2. *-commutative 31.2%

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

   7. Simplified 31.2%

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

      1. pow1 31.2%

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

      2. sqrt-unprod 60.4%

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

      3. *-commutative 60.4%

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

      4. associate-*l* 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. unpow1 68.2%

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

       2. associate-*r* 74.4%

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

   11. Simplified 74.4%

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

   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 31.5%

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

   4. Taylor expanded in U around 0 28.9%

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

   5. Taylor expanded in l around 0 29.4%

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

      1. associate-/l* 29.4%

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

      2. unpow2 29.4%

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

      3. sub-neg 29.4%

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

      4. associate-/l* 29.4%

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

      5. metadata-eval 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in t around 0 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. associate-/l* 44.4%

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

      3. *-commutative 44.4%

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

      4. sub-neg 44.4%

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

      5. associate-/l* 41.9%

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

      6. metadata-eval 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. *-lft-identity 41.9%

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

       2. associate-/r/ 44.5%

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

       3. +-commutative 44.5%

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

       4. associate-/l* 47.0%

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

   12. Simplified 47.0%

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

4. Final simplification 65.4%

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

Alternative 3: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.2e-250)
   (sqrt (* (* 2.0 n) (* U t)))
   (if (<= l 3.2e-244)
     (* (sqrt (* 2.0 (* n U))) (sqrt t))
     (if (<= l 8e-7)
       (sqrt
        (*
         (* (* 2.0 n) U)
         (+ t (* (/ l Om) (fma l -2.0 (/ (* n (* l U*)) Om))))))
       (if (<= l 6.8e+180)
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
         (*
          (* l (sqrt 2.0))
          (sqrt (* (/ n Om) (* U (+ -2.0 (/ (* n U*) Om)))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.2e-250) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 3.2e-244) {
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	} else if (l <= 8e-7) {
		tmp = sqrt((((2.0 * n) * U) * (t + ((l / Om) * fma(l, -2.0, ((n * (l * U_42_)) / Om))))));
	} else if (l <= 6.8e+180) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U)))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.2e-250)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (l <= 3.2e-244)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	elseif (l <= 8e-7)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(l * U_42_)) / Om))))));
	elseif (l <= 6.8e+180)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n / Om) * Float64(U * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.2e-250], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.2e-244], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e-7], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.8e+180], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 7.19999999999999964e-250

   1. Initial program 51.9%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in t around inf 40.8%

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

   if 7.19999999999999964e-250 < l < 3.1999999999999998e-244

   1. Initial program 23.8%

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

   2. Taylor expanded in t around inf 23.8%

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

      1. sqrt-prod 60.0%

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

      2. associate-*l* 60.0%

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

   4. Applied egg-rr 60.0%

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

   if 3.1999999999999998e-244 < l < 7.9999999999999996e-7

   1. Initial program 65.1%

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

   3. Simplified 60.5%

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

      1. sqrt-prod 41.5%

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

   5. Applied egg-rr 41.5%

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

      1. *-commutative 41.5%

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

      2. *-commutative 41.5%

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

   7. Simplified 41.5%

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

      1. pow1 41.5%

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

      2. sqrt-unprod 60.5%

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

      3. *-commutative 60.5%

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

      4. associate-*l* 65.9%

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

   9. Applied egg-rr 65.9%

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

       1. unpow1 65.9%

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

       2. associate-*r* 73.4%

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

   11. Simplified 73.4%

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

   12. Taylor expanded in U* around inf 69.7%

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

   if 7.9999999999999996e-7 < l < 6.79999999999999969e180

   1. Initial program 53.8%

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

   3. Simplified 66.6%

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

   if 6.79999999999999969e180 < l

   1. Initial program 6.0%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in U around 0 23.2%

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

   5. Taylor expanded in l around 0 23.7%

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

      1. associate-/l* 23.7%

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

      2. unpow2 23.7%

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

      3. sub-neg 23.7%

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

      4. associate-/l* 23.7%

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

      5. metadata-eval 23.7%

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

   7. Simplified 23.7%

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

   8. Taylor expanded in t around 0 86.9%

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

      1. *-un-lft-identity 86.9%

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

      2. associate-/l* 79.4%

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

      3. *-commutative 79.4%

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

      4. sub-neg 79.4%

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

      5. associate-/l* 79.4%

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

      6. metadata-eval 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. *-lft-identity 79.4%

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

       2. associate-/r/ 76.6%

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

       3. +-commutative 76.6%

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

       4. associate-/l* 76.6%

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

   12. Simplified 76.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 4: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 n)
           (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))))
   (if (<= l 7.2e-250)
     (sqrt (* (* 2.0 n) (* U t)))
     (if (<= l 1.7e-243)
       (* (sqrt (* 2.0 (* n U))) (sqrt t))
       (if (<= l 4.4e-182)
         t_1
         (if (<= l 1.1e-95)
           (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (/ l (/ Om l))))))
           (if (<= l 1.1e+59)
             t_1
             (*
              (* l (sqrt 2.0))
              (sqrt (* (/ n Om) (* U (+ -2.0 (/ (* n U*) Om)))))))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	double tmp;
	if (l <= 7.2e-250) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 1.7e-243) {
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	} else if (l <= 4.4e-182) {
		tmp = t_1;
	} else if (l <= 1.1e-95) {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	} else if (l <= 1.1e+59) {
		tmp = t_1;
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    if (l <= 7.2d-250) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else if (l <= 1.7d-243) then
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(t)
    else if (l <= 4.4d-182) then
        tmp = t_1
    else if (l <= 1.1d-95) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l / (om / l))))))
    else if (l <= 1.1d+59) then
        tmp = t_1
    else
        tmp = (l * sqrt(2.0d0)) * sqrt(((n / om) * (u * ((-2.0d0) + ((n * u_42) / om)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	double tmp;
	if (l <= 7.2e-250) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 1.7e-243) {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	} else if (l <= 4.4e-182) {
		tmp = t_1;
	} else if (l <= 1.1e-95) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	} else if (l <= 1.1e+59) {
		tmp = t_1;
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	tmp = 0
	if l <= 7.2e-250:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	elif l <= 1.7e-243:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	elif l <= 4.4e-182:
		tmp = t_1
	elif l <= 1.1e-95:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))))
	elif l <= 1.1e+59:
		tmp = t_1
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))))
	tmp = 0.0
	if (l <= 7.2e-250)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (l <= 1.7e-243)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	elseif (l <= 4.4e-182)
		tmp = t_1;
	elseif (l <= 1.1e-95)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))));
	elseif (l <= 1.1e+59)
		tmp = t_1;
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n / Om) * Float64(U * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	tmp = 0.0;
	if (l <= 7.2e-250)
		tmp = sqrt(((2.0 * n) * (U * t)));
	elseif (l <= 1.7e-243)
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	elseif (l <= 4.4e-182)
		tmp = t_1;
	elseif (l <= 1.1e-95)
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	elseif (l <= 1.1e+59)
		tmp = t_1;
	else
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 7.2e-250], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.7e-243], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.4e-182], t$95$1, If[LessEqual[l, 1.1e-95], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.1e+59], t$95$1, N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 7.19999999999999964e-250

   1. Initial program 51.9%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in t around inf 40.8%

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

   if 7.19999999999999964e-250 < l < 1.69999999999999998e-243

   1. Initial program 23.8%

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

   2. Taylor expanded in t around inf 23.8%

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

      1. sqrt-prod 60.0%

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

      2. associate-*l* 60.0%

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

   4. Applied egg-rr 60.0%

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

   if 1.69999999999999998e-243 < l < 4.3999999999999999e-182 or 1.0999999999999999e-95 < l < 1.1e59

   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 64.8%

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

   4. Taylor expanded in U around 0 65.9%

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

   if 4.3999999999999999e-182 < l < 1.0999999999999999e-95

   1. Initial program 53.8%

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

   2. Taylor expanded in Om around inf 50.6%

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

      1. unpow2 50.6%

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

      2. associate-/l* 50.6%

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

   4. Simplified 50.6%

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

   if 1.1e59 < l

   1. Initial program 24.6%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in U around 0 39.0%

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

   5. Taylor expanded in l around 0 39.3%

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

      1. associate-/l* 39.3%

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

      2. unpow2 39.3%

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

      3. sub-neg 39.3%

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

      4. associate-/l* 39.3%

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

      5. metadata-eval 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in t around 0 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. associate-/l* 70.4%

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

      3. *-commutative 70.4%

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

      4. sub-neg 70.4%

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

      5. associate-/l* 72.4%

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

      6. metadata-eval 72.4%

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

   10. Applied egg-rr 72.4%

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

       1. *-lft-identity 72.4%

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

       2. associate-/r/ 72.8%

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

       3. +-commutative 72.8%

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

       4. associate-/l* 70.9%

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

   12. Simplified 70.9%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 5: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\\ \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, t_1\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + t_1\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \end{array} \]

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)))
   (if (<= l 6.6e-250)
     (sqrt (* (* 2.0 n) (* U t)))
     (if (<= l 8.6e-245)
       (* (sqrt (* 2.0 (* n U))) (sqrt t))
       (if (<= l 5.4e-6)
         (sqrt (* (* (* 2.0 n) U) (+ t (* (/ l Om) (fma l -2.0 t_1)))))
         (if (<= l 2.6e+144)
           (sqrt
            (*
             (* 2.0 n)
             (* U (+ t (/ (* l l) (/ Om (+ -2.0 (/ n (/ Om U*)))))))))
           (if (<= l 6e+180)
             (sqrt (* 2.0 (/ (* n (* l (* U (+ (* l -2.0) t_1)))) Om)))
             (*
              (* l (sqrt 2.0))
              (sqrt (* (/ n Om) (* U (+ -2.0 (/ (* n U*) 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;
	double tmp;
	if (l <= 6.6e-250) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (l <= 8.6e-245) {
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	} else if (l <= 5.4e-6) {
		tmp = sqrt((((2.0 * n) * U) * (t + ((l / Om) * fma(l, -2.0, t_1)))));
	} else if (l <= 2.6e+144) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	} else if (l <= 6e+180) {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + t_1)))) / Om)));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n / Om) * (U * (-2.0 + ((n * U_42_) / Om)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * Float64(l * U_42_)) / Om)
	tmp = 0.0
	if (l <= 6.6e-250)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (l <= 8.6e-245)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	elseif (l <= 5.4e-6)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, t_1)))));
	elseif (l <= 2.6e+144)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * l) / Float64(Om / Float64(-2.0 + Float64(n / Float64(Om / U_42_)))))))));
	elseif (l <= 6e+180)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(l * -2.0) + t_1)))) / Om)));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n / Om) * Float64(U * Float64(-2.0 + Float64(Float64(n * U_42_) / 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[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, 6.6e-250], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.6e-245], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.4e-6], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.6e+144], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] / N[(Om / N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6e+180], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * N[(N[(l * -2.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n / Om), $MachinePrecision] * N[(U * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < 6.6e-250

   1. Initial program 51.9%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in t around inf 40.8%

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

   if 6.6e-250 < l < 8.60000000000000005e-245

   1. Initial program 23.8%

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

   2. Taylor expanded in t around inf 23.8%

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

      1. sqrt-prod 60.0%

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

      2. associate-*l* 60.0%

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

   4. Applied egg-rr 60.0%

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

   if 8.60000000000000005e-245 < l < 5.39999999999999997e-6

   1. Initial program 65.7%

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

   3. Simplified 61.2%

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

      1. sqrt-prod 42.5%

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

   5. Applied egg-rr 42.5%

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

      1. *-commutative 42.5%

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

      2. *-commutative 42.5%

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

   7. Simplified 42.5%

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

      1. pow1 42.5%

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

      2. sqrt-unprod 61.2%

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

      3. *-commutative 61.2%

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

      4. associate-*l* 66.5%

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

   9. Applied egg-rr 66.5%

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

       1. unpow1 66.5%

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

       2. associate-*r* 73.8%

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

   11. Simplified 73.8%

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

   12. Taylor expanded in U* around inf 70.2%

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

   if 5.39999999999999997e-6 < l < 2.5999999999999999e144

   1. Initial program 56.1%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in U around 0 59.1%

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

   5. Taylor expanded in l around 0 59.1%

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

      1. associate-/l* 59.1%

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

      2. unpow2 59.1%

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

      3. sub-neg 59.1%

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

      4. associate-/l* 62.5%

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

      5. metadata-eval 62.5%

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

   7. Simplified 62.5%

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

   if 2.5999999999999999e144 < l < 6.00000000000000006e180

   1. Initial program 34.5%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in U around 0 35.1%

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

   5. Taylor expanded in t around 0 100.0%

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

   if 6.00000000000000006e180 < l

   1. Initial program 6.0%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in U around 0 23.2%

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

   5. Taylor expanded in l around 0 23.7%

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

      1. associate-/l* 23.7%

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

      2. unpow2 23.7%

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

      3. sub-neg 23.7%

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

      4. associate-/l* 23.7%

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

      5. metadata-eval 23.7%

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

   7. Simplified 23.7%

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

   8. Taylor expanded in t around 0 86.9%

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

      1. *-un-lft-identity 86.9%

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

      2. associate-/l* 79.4%

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

      3. *-commutative 79.4%

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

      4. sub-neg 79.4%

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

      5. associate-/l* 79.4%

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

      6. metadata-eval 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. *-lft-identity 79.4%

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

       2. associate-/r/ 76.6%

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

       3. +-commutative 76.6%

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

       4. associate-/l* 76.6%

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

   12. Simplified 76.6%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n}{Om} \cdot \left(U \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)\right)}\\ \end{array} \]

Alternative 6: 51.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{if}\;Om \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 2.55 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;Om \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ l (/ Om l)))))
        (t_2
         (sqrt
          (*
           2.0
           (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l U*)) Om))))) Om)))))
   (if (<= Om -2.65e-129)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= Om 2.55e-154)
       t_2
       (if (<= Om 1.35e-45)
         (sqrt (* (* 2.0 n) (* U (+ t (* (/ (* n (* l l)) Om) (/ U* Om))))))
         (if (<= Om 2.1e-21) t_2 (sqrt (* (* (* 2.0 n) U) t_1))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double t_2 = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	double tmp;
	if (Om <= -2.65e-129) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 2.55e-154) {
		tmp = t_2;
	} else if (Om <= 1.35e-45) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else if (Om <= 2.1e-21) {
		tmp = t_2;
	} else {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l / (om / l)))
    t_2 = sqrt((2.0d0 * ((n * (l * (u * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))))) / om)))
    if (om <= (-2.65d-129)) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (om <= 2.55d-154) then
        tmp = t_2
    else if (om <= 1.35d-45) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((n * (l * l)) / om) * (u_42 / om))))))
    else if (om <= 2.1d-21) then
        tmp = t_2
    else
        tmp = sqrt((((2.0d0 * n) * u) * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double t_2 = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	double tmp;
	if (Om <= -2.65e-129) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 2.55e-154) {
		tmp = t_2;
	} else if (Om <= 1.35e-45) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else if (Om <= 2.1e-21) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l / (Om / l)))
	t_2 = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	tmp = 0
	if Om <= -2.65e-129:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif Om <= 2.55e-154:
		tmp = t_2
	elif Om <= 1.35e-45:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))))
	elif Om <= 2.1e-21:
		tmp = t_2
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))
	t_2 = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))))) / Om)))
	tmp = 0.0
	if (Om <= -2.65e-129)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (Om <= 2.55e-154)
		tmp = t_2;
	elseif (Om <= 1.35e-45)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(n * Float64(l * l)) / Om) * Float64(U_42_ / Om))))));
	elseif (Om <= 2.1e-21)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l / (Om / l)));
	t_2 = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	tmp = 0.0;
	if (Om <= -2.65e-129)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (Om <= 2.55e-154)
		tmp = t_2;
	elseif (Om <= 1.35e-45)
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	elseif (Om <= 2.1e-21)
		tmp = t_2;
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -2.65e-129], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.55e-154], t$95$2, If[LessEqual[Om, 1.35e-45], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.1e-21], t$95$2, N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -2.64999999999999987e-129

   1. Initial program 49.7%

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

   3. Simplified 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 n around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. unpow2 51.6%

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

      3. associate-/l* 58.1%

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

   6. Simplified 58.1%

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

   if -2.64999999999999987e-129 < Om < 2.5499999999999999e-154 or 1.34999999999999992e-45 < Om < 2.10000000000000013e-21

   1. Initial program 36.0%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in U around 0 55.5%

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

   5. Taylor expanded in t around 0 56.4%

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

   if 2.5499999999999999e-154 < Om < 1.34999999999999992e-45

   1. Initial program 67.5%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in U around 0 64.4%

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

   5. Taylor expanded in l around 0 64.2%

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

      1. associate-/l* 54.4%

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

      2. unpow2 54.4%

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

      3. sub-neg 54.4%

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

      4. associate-/l* 54.2%

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

      5. metadata-eval 54.2%

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

   7. Simplified 54.2%

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

   8. Taylor expanded in Om around 0 62.3%

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

      1. associate-*r* 62.2%

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

      2. unpow2 62.2%

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

      3. times-frac 61.9%

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

      4. unpow2 61.9%

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

   10. Simplified 61.9%

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

   if 2.10000000000000013e-21 < Om

   1. Initial program 56.1%

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

   2. Taylor expanded in Om around inf 56.5%

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

      1. unpow2 56.5%

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

      2. associate-/l* 59.1%

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

   4. Simplified 59.1%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.55 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 7: 51.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -1.24 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 1.75 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot -2 - \frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ l (/ Om l))))))
   (if (<= Om -1.24e-129)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= Om 1.75e-154)
       (sqrt
        (* 2.0 (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l U*)) Om))))) Om)))
       (if (<= Om 4e-40)
         (sqrt (* (* 2.0 n) (* U (+ t (* (/ (* n (* l l)) Om) (/ U* Om))))))
         (if (<= Om 3.5e-12)
           (sqrt
            (*
             2.0
             (/ (* (* n l) (* U (- (* l -2.0) (* (/ n Om) (* U l))))) Om)))
           (sqrt (* (* (* 2.0 n) U) t_1))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -1.24e-129) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.75e-154) {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	} else if (Om <= 4e-40) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else if (Om <= 3.5e-12) {
		tmp = sqrt((2.0 * (((n * l) * (U * ((l * -2.0) - ((n / Om) * (U * l))))) / Om)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l / (om / l)))
    if (om <= (-1.24d-129)) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (om <= 1.75d-154) then
        tmp = sqrt((2.0d0 * ((n * (l * (u * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))))) / om)))
    else if (om <= 4d-40) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((n * (l * l)) / om) * (u_42 / om))))))
    else if (om <= 3.5d-12) then
        tmp = sqrt((2.0d0 * (((n * l) * (u * ((l * (-2.0d0)) - ((n / om) * (u * l))))) / om)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -1.24e-129) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.75e-154) {
		tmp = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	} else if (Om <= 4e-40) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else if (Om <= 3.5e-12) {
		tmp = Math.sqrt((2.0 * (((n * l) * (U * ((l * -2.0) - ((n / Om) * (U * l))))) / Om)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l / (Om / l)))
	tmp = 0
	if Om <= -1.24e-129:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif Om <= 1.75e-154:
		tmp = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	elif Om <= 4e-40:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))))
	elif Om <= 3.5e-12:
		tmp = math.sqrt((2.0 * (((n * l) * (U * ((l * -2.0) - ((n / Om) * (U * l))))) / Om)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))
	tmp = 0.0
	if (Om <= -1.24e-129)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (Om <= 1.75e-154)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))))) / Om)));
	elseif (Om <= 4e-40)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(n * Float64(l * l)) / Om) * Float64(U_42_ / Om))))));
	elseif (Om <= 3.5e-12)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(n * l) * Float64(U * Float64(Float64(l * -2.0) - Float64(Float64(n / Om) * Float64(U * l))))) / Om)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l / (Om / l)));
	tmp = 0.0;
	if (Om <= -1.24e-129)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (Om <= 1.75e-154)
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	elseif (Om <= 4e-40)
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	elseif (Om <= 3.5e-12)
		tmp = sqrt((2.0 * (((n * l) * (U * ((l * -2.0) - ((n / Om) * (U * l))))) / Om)));
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.24e-129], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.75e-154], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 4e-40], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 3.5e-12], N[Sqrt[N[(2.0 * N[(N[(N[(n * l), $MachinePrecision] * N[(U * N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n / Om), $MachinePrecision] * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if Om < -1.24000000000000004e-129

   1. Initial program 49.7%

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

   3. Simplified 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 n around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. unpow2 51.6%

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

      3. associate-/l* 58.1%

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

   6. Simplified 58.1%

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

   if -1.24000000000000004e-129 < Om < 1.75e-154

   1. Initial program 37.3%

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

   3. Simplified 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 U around 0 58.5%

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

   5. Taylor expanded in t around 0 55.8%

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

   if 1.75e-154 < Om < 3.9999999999999997e-40

   1. Initial program 66.3%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in U around 0 63.5%

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

   5. Taylor expanded in l around 0 63.3%

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

      1. associate-/l* 54.4%

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

      2. unpow2 54.4%

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

      3. sub-neg 54.4%

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

      4. associate-/l* 54.2%

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

      5. metadata-eval 54.2%

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

   7. Simplified 54.2%

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

   8. Taylor expanded in Om around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. unpow2 61.5%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

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

   10. Simplified 61.3%

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

   if 3.9999999999999997e-40 < Om < 3.5e-12

   1. Initial program 43.4%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in U* around 0 17.5%

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

      1. associate-*r* 45.0%

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

      2. +-commutative 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in t around 0 44.6%

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

      1. associate-*r* 85.3%

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

      2. *-commutative 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-*l/ 85.3%

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

      5. *-commutative 85.3%

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

   9. Simplified 85.3%

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

   if 3.5e-12 < Om

   1. Initial program 55.0%

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

   2. Taylor expanded in Om around inf 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/l* 58.1%

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

   4. Simplified 58.1%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.24 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.75 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(n \cdot \ell\right) \cdot \left(U \cdot \left(\ell \cdot -2 - \frac{n}{Om} \cdot \left(U \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 8: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -5.2 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -5.2e+77)
   (sqrt
    (* 2.0 (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ n (/ Om (* U l)))))))))))
   (if (<= Om 1.35e-17)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (/ l (/ Om l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -5.2e+77) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	} else if (Om <= 1.35e-17) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-5.2d+77)) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - (n / (om / (u * l)))))))))))
    else if (om <= 1.35d-17) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l / (om / l))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -5.2e+77) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	} else if (Om <= 1.35e-17) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -5.2e+77:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))))
	elif Om <= 1.35e-17:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -5.2e+77)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(n / Float64(Om / Float64(U * l)))))))))));
	elseif (Om <= 1.35e-17)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -5.2e+77)
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	elseif (Om <= 1.35e-17)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -5.2e+77], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(n / N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.35e-17], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -5.2000000000000004e77

   1. Initial program 56.1%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in U* around 0 46.6%

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

      1. associate-*r* 52.5%

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

      2. +-commutative 52.5%

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

   6. Simplified 68.0%

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

   if -5.2000000000000004e77 < Om < 1.3500000000000001e-17

   1. Initial program 41.9%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in U around 0 55.6%

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

   if 1.3500000000000001e-17 < Om

   1. Initial program 56.1%

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

   2. Taylor expanded in Om around inf 56.5%

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

      1. unpow2 56.5%

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

      2. associate-/l* 59.1%

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

   4. Simplified 59.1%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -5.2 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 9: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.6e-116)
   (sqrt (* 2.0 (* U (* n (+ t (* (/ (* l l) Om) -2.0))))))
   (if (<= l 2.8e+139)
     (sqrt
      (* (* 2.0 n) (* U (+ t (/ (* l l) (/ Om (+ -2.0 (/ n (/ Om U*)))))))))
     (sqrt
      (* 2.0 (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l 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 <= 2.6e-116) {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else if (l <= 2.8e+139) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	} else {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * 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 (l <= 2.6d-116) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) / om) * (-2.0d0)))))))
    else if (l <= 2.8d+139) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * l) / (om / ((-2.0d0) + (n / (om / u_42)))))))))
    else
        tmp = sqrt((2.0d0 * ((n * (l * (u * ((l * (-2.0d0)) + ((n * (l * 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 (l <= 2.6e-116) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else if (l <= 2.8e+139) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	} else {
		tmp = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.6e-116:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))))
	elif l <= 2.8e+139:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))))
	else:
		tmp = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.6e-116)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))));
	elseif (l <= 2.8e+139)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * l) / Float64(Om / Float64(-2.0 + Float64(n / Float64(Om / U_42_)))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * 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 (l <= 2.6e-116)
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	elseif (l <= 2.8e+139)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * l) / (Om / (-2.0 + (n / (Om / U_42_)))))))));
	else
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * 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[l, 2.6e-116], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.8e+139], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * l), $MachinePrecision] / N[(Om / N[(-2.0 + N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.6e-116

   1. Initial program 54.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in U around 0 54.7%

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

   5. Taylor expanded in n around 0 50.1%

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

      1. associate-*r* 52.4%

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

      2. unpow2 52.4%

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

   7. Simplified 52.4%

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

   if 2.6e-116 < l < 2.7999999999999998e139

   1. Initial program 57.6%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in U around 0 59.7%

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

   5. Taylor expanded in l around 0 57.9%

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

      1. associate-/l* 52.4%

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

      2. unpow2 52.4%

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

      3. sub-neg 52.4%

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

      4. associate-/l* 57.6%

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

      5. metadata-eval 57.6%

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

   7. Simplified 57.6%

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

   if 2.7999999999999998e139 < l

   1. Initial program 11.1%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in U around 0 27.7%

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

   5. Taylor expanded in t around 0 48.1%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \ell}{\frac{Om}{-2 + \frac{n}{\frac{Om}{U*}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 10: 50.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -4.5e-79)
   (sqrt
    (* 2.0 (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ n (/ Om (* U l)))))))))))
   (if (<= Om 3.3e-19)
     (sqrt
      (* 2.0 (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l U*)) Om))))) Om)))
     (sqrt (* (* (* 2.0 n) U) (+ t (* -2.0 (/ l (/ Om l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -4.5e-79) {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	} else if (Om <= 3.3e-19) {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-4.5d-79)) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - (n / (om / (u * l)))))))))))
    else if (om <= 3.3d-19) then
        tmp = sqrt((2.0d0 * ((n * (l * (u * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))))) / om)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((-2.0d0) * (l / (om / l))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -4.5e-79) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	} else if (Om <= 3.3e-19) {
		tmp = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -4.5e-79:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))))
	elif Om <= 3.3e-19:
		tmp = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -4.5e-79)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(n / Float64(Om / Float64(U * l)))))))))));
	elseif (Om <= 3.3e-19)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * Float64(l * Float64(U * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))))) / Om)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -4.5e-79)
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	elseif (Om <= 3.3e-19)
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	else
		tmp = sqrt((((2.0 * n) * U) * (t + (-2.0 * (l / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -4.5e-79], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(l / N[(Om / N[(N[(l * -2.0), $MachinePrecision] - N[(n / N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 3.3e-19], N[Sqrt[N[(2.0 * N[(N[(n * N[(l * N[(U * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -4.5000000000000003e-79

   1. Initial program 50.7%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in U* around 0 45.6%

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

      1. associate-*r* 48.8%

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

      2. +-commutative 48.8%

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

   6. Simplified 59.7%

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

   if -4.5000000000000003e-79 < Om < 3.2999999999999998e-19

   1. Initial program 43.3%

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

   3. Simplified 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 58.1%

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

   5. Taylor expanded in t around 0 51.0%

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

   if 3.2999999999999998e-19 < Om

   1. Initial program 56.1%

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

   2. Taylor expanded in Om around inf 56.5%

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

      1. unpow2 56.5%

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

      2. associate-/l* 59.1%

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

   4. Simplified 59.1%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 11: 49.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -4.4 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ l (/ Om l))))))
   (if (<= Om -4.4e-133)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= Om 1.6e-101)
       (sqrt (* -2.0 (/ (* (* n (* l l)) (* U (- 2.0 (/ n (/ Om U*))))) Om)))
       (sqrt (* (* (* 2.0 n) U) t_1))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -4.4e-133) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.6e-101) {
		tmp = sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l / (om / l)))
    if (om <= (-4.4d-133)) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (om <= 1.6d-101) then
        tmp = sqrt(((-2.0d0) * (((n * (l * l)) * (u * (2.0d0 - (n / (om / u_42))))) / om)))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -4.4e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.6e-101) {
		tmp = Math.sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l / (Om / l)))
	tmp = 0
	if Om <= -4.4e-133:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif Om <= 1.6e-101:
		tmp = math.sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))
	tmp = 0.0
	if (Om <= -4.4e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (Om <= 1.6e-101)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(n * Float64(l * l)) * Float64(U * Float64(2.0 - Float64(n / Float64(Om / U_42_))))) / Om)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l / (Om / l)));
	tmp = 0.0;
	if (Om <= -4.4e-133)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (Om <= 1.6e-101)
		tmp = sqrt((-2.0 * (((n * (l * l)) * (U * (2.0 - (n / (Om / U_42_))))) / Om)));
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -4.4e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.6e-101], N[Sqrt[N[(-2.0 * N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(n / N[(Om / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -4.4000000000000001e-133

   1. Initial program 48.9%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in n around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. associate-/l* 57.0%

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

   6. Simplified 57.0%

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

   if -4.4000000000000001e-133 < Om < 1.59999999999999989e-101

   1. Initial program 44.0%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in U around 0 62.3%

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

   5. Taylor expanded in l around -inf 48.0%

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

      1. associate-*r* 47.5%

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

      2. unpow2 47.5%

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

      3. *-commutative 47.5%

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

      4. mul-1-neg 47.5%

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

      5. unsub-neg 47.5%

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

      6. associate-/l* 42.2%

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

   7. Simplified 42.2%

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

   if 1.59999999999999989e-101 < Om

   1. Initial program 54.7%

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

   2. Taylor expanded in Om around inf 52.9%

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

      1. unpow2 52.9%

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

      2. associate-/l* 55.0%

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

   4. Simplified 55.0%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.4 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n}{\frac{Om}{U*}}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 12: 49.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ l (/ Om l))))))
   (if (<= Om -9.5e-133)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= Om 1.3e-99)
       (sqrt (* 2.0 (/ n (/ Om (* (* l l) (* U (+ -2.0 (* U* (/ n Om)))))))))
       (sqrt (* (* (* 2.0 n) U) t_1))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -9.5e-133) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.3e-99) {
		tmp = sqrt((2.0 * (n / (Om / ((l * l) * (U * (-2.0 + (U_42_ * (n / Om)))))))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l / (om / l)))
    if (om <= (-9.5d-133)) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (om <= 1.3d-99) then
        tmp = sqrt((2.0d0 * (n / (om / ((l * l) * (u * ((-2.0d0) + (u_42 * (n / om)))))))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -9.5e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.3e-99) {
		tmp = Math.sqrt((2.0 * (n / (Om / ((l * l) * (U * (-2.0 + (U_42_ * (n / Om)))))))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l / (Om / l)))
	tmp = 0
	if Om <= -9.5e-133:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif Om <= 1.3e-99:
		tmp = math.sqrt((2.0 * (n / (Om / ((l * l) * (U * (-2.0 + (U_42_ * (n / Om)))))))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))
	tmp = 0.0
	if (Om <= -9.5e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (Om <= 1.3e-99)
		tmp = sqrt(Float64(2.0 * Float64(n / Float64(Om / Float64(Float64(l * l) * Float64(U * Float64(-2.0 + Float64(U_42_ * Float64(n / Om)))))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l / (Om / l)));
	tmp = 0.0;
	if (Om <= -9.5e-133)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (Om <= 1.3e-99)
		tmp = sqrt((2.0 * (n / (Om / ((l * l) * (U * (-2.0 + (U_42_ * (n / Om)))))))));
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -9.5e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.3e-99], N[Sqrt[N[(2.0 * N[(n / N[(Om / N[(N[(l * l), $MachinePrecision] * N[(U * N[(-2.0 + N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -9.4999999999999992e-133

   1. Initial program 48.9%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in n around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. associate-/l* 57.0%

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

   6. Simplified 57.0%

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

   if -9.4999999999999992e-133 < Om < 1.30000000000000003e-99

   1. Initial program 44.0%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in U around 0 62.3%

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

   5. Taylor expanded in l around 0 54.9%

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

      1. associate-/l* 47.5%

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

      2. unpow2 47.5%

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

      3. sub-neg 47.5%

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

      4. associate-/l* 45.7%

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

      5. metadata-eval 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in t around 0 48.0%

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

      1. associate-/l* 49.2%

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

      2. unpow2 49.2%

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

      3. *-commutative 49.2%

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

      4. sub-neg 49.2%

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

      5. associate-*l/ 47.5%

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

      6. metadata-eval 47.5%

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

      7. +-commutative 47.5%

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

      8. *-commutative 47.5%

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

   10. Simplified 47.5%

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

   if 1.30000000000000003e-99 < Om

   1. Initial program 54.7%

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

   2. Taylor expanded in Om around inf 52.9%

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

      1. unpow2 52.9%

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

      2. associate-/l* 55.0%

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

   4. Simplified 55.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 13: 50.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq -9.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 3.8 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ l (/ Om l))))))
   (if (<= Om -9.8e-133)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= Om 3.8e-40)
       (sqrt (* (* 2.0 n) (* U (+ t (* (/ (* n (* l l)) Om) (/ U* Om))))))
       (sqrt (* (* (* 2.0 n) U) t_1))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -9.8e-133) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 3.8e-40) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l / (om / l)))
    if (om <= (-9.8d-133)) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (om <= 3.8d-40) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((n * (l * l)) / om) * (u_42 / om))))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= -9.8e-133) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 3.8e-40) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l / (Om / l)))
	tmp = 0
	if Om <= -9.8e-133:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif Om <= 3.8e-40:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))
	tmp = 0.0
	if (Om <= -9.8e-133)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (Om <= 3.8e-40)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(n * Float64(l * l)) / Om) * Float64(U_42_ / Om))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l / (Om / l)));
	tmp = 0.0;
	if (Om <= -9.8e-133)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (Om <= 3.8e-40)
		tmp = sqrt(((2.0 * n) * (U * (t + (((n * (l * l)) / Om) * (U_42_ / Om))))));
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -9.8e-133], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 3.8e-40], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -9.79999999999999992e-133

   1. Initial program 48.9%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in n around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. associate-/l* 57.0%

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

   6. Simplified 57.0%

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

   if -9.79999999999999992e-133 < Om < 3.7999999999999999e-40

   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 63.2%

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

   4. Taylor expanded in U around 0 61.8%

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

   5. Taylor expanded in l around 0 55.9%

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

      1. associate-/l* 48.5%

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

      2. unpow2 48.5%

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

      3. sub-neg 48.5%

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

      4. associate-/l* 47.0%

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

      5. metadata-eval 47.0%

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

   7. Simplified 47.0%

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

   8. Taylor expanded in Om around 0 45.0%

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

      1. associate-*r* 44.5%

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

      2. unpow2 44.5%

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

      3. times-frac 52.5%

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

      4. unpow2 52.5%

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

   10. Simplified 52.5%

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

   if 3.7999999999999999e-40 < Om

   1. Initial program 54.0%

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

   2. Taylor expanded in Om around inf 54.4%

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

      1. unpow2 54.4%

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

      2. associate-/l* 56.9%

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

   4. Simplified 56.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9.8 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.8 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{n \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{U*}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 14: 47.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\\ \mathbf{if}\;Om \leq 6 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t_1\right)}\\ \mathbf{elif}\;Om \leq 1.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ l (/ Om l))))))
   (if (<= Om 6e-298)
     (sqrt (* (* 2.0 n) (* U t_1)))
     (if (<= Om 1.1e-191)
       (/ (sqrt (* (- U* U) (* 2.0 U))) (/ Om (* n l)))
       (sqrt (* (* (* 2.0 n) U) t_1))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= 6e-298) {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.1e-191) {
		tmp = sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	} else {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + ((-2.0d0) * (l / (om / l)))
    if (om <= 6d-298) then
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    else if (om <= 1.1d-191) then
        tmp = sqrt(((u_42 - u) * (2.0d0 * u))) / (om / (n * l))
    else
        tmp = sqrt((((2.0d0 * n) * u) * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (l / (Om / l)));
	double tmp;
	if (Om <= 6e-298) {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	} else if (Om <= 1.1e-191) {
		tmp = Math.sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	t_1 = t + (-2.0 * (l / (Om / l)))
	tmp = 0
	if Om <= 6e-298:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	elif Om <= 1.1e-191:
		tmp = math.sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l))))
	tmp = 0.0
	if (Om <= 6e-298)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	elseif (Om <= 1.1e-191)
		tmp = Float64(sqrt(Float64(Float64(U_42_ - U) * Float64(2.0 * U))) / Float64(Om / Float64(n * l)));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + (-2.0 * (l / (Om / l)));
	tmp = 0.0;
	if (Om <= 6e-298)
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	elseif (Om <= 1.1e-191)
		tmp = sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, 6e-298], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.1e-191], N[(N[Sqrt[N[(N[(U$42$ - U), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Om < 5.9999999999999999e-298

   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 55.7%

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

   4. Taylor expanded in n around 0 47.2%

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

      1. *-commutative 47.2%

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

      2. unpow2 47.2%

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

      3. associate-/l* 52.1%

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

   6. Simplified 52.1%

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

   if 5.9999999999999999e-298 < Om < 1.09999999999999999e-191

   1. Initial program 20.7%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in n around inf 46.4%

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

      1. associate-/l* 46.4%

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

      2. *-commutative 46.4%

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

   6. Simplified 46.4%

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

      1. associate-*l/ 46.5%

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

   8. Applied egg-rr 46.5%

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

      1. div-inv 46.4%

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

      2. sqrt-unprod 46.7%

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

      3. associate-/r* 46.7%

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

   10. Applied egg-rr 46.7%

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

       1. associate-*r/ 46.7%

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

       2. *-rgt-identity 46.7%

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

       3. associate-*r* 46.7%

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

       4. associate-/l/ 46.7%

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

       5. *-commutative 46.7%

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

   12. Simplified 46.7%

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

   if 1.09999999999999999e-191 < Om

   1. Initial program 56.1%

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

   2. Taylor expanded in Om around inf 51.8%

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

      1. unpow2 51.8%

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

      2. associate-/l* 53.7%

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

   4. Simplified 53.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 6 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 1.1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}}{\frac{Om}{n \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\\ \end{array} \]

Alternative 15: 45.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7.8e+197)
   (sqrt (* 2.0 (* U (* n (+ t (* (/ (* l l) Om) -2.0))))))
   (/ (sqrt (* (- U* U) (* 2.0 U))) (/ Om (* n l)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.8e+197) {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else {
		tmp = sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 7.8d+197) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) / om) * (-2.0d0)))))))
    else
        tmp = sqrt(((u_42 - u) * (2.0d0 * u))) / (om / (n * l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7.8e+197) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else {
		tmp = Math.sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7.8e+197:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))))
	else:
		tmp = math.sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7.8e+197)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))));
	else
		tmp = Float64(sqrt(Float64(Float64(U_42_ - U) * Float64(2.0 * U))) / Float64(Om / Float64(n * l)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 7.8e+197)
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	else
		tmp = sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7.8e+197], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U$42$ - U), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.8e197

   1. Initial program 54.4%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in U around 0 55.2%

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

   5. Taylor expanded in n around 0 49.1%

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

      1. associate-*r* 50.3%

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

      2. unpow2 50.3%

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

   7. Simplified 50.3%

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

   if 7.8e197 < l

   1. Initial program 6.5%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in n around inf 21.6%

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

      1. associate-/l* 21.6%

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

      2. *-commutative 21.6%

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

   6. Simplified 21.6%

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

      1. associate-*l/ 21.6%

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

   8. Applied egg-rr 21.6%

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

      1. div-inv 21.6%

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

      2. sqrt-unprod 21.6%

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

      3. associate-/r* 25.1%

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

   10. Applied egg-rr 25.1%

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

       1. associate-*r/ 25.1%

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

       2. *-rgt-identity 25.1%

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

       3. associate-*r* 25.1%

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

       4. associate-/l/ 21.6%

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

       5. *-commutative 21.6%

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

   12. Simplified 21.6%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 16: 48.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq -2.25 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -2.25e+91)
   (sqrt (* 2.0 (* U (* n (+ t (* (/ (* l l) Om) -2.0))))))
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (/ l (/ Om l)))))))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -2.25e+91) {
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l / (Om / l)))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-2.25d+91)) then
        tmp = sqrt((2.0d0 * (u * (n * (t + (((l * l) / om) * (-2.0d0)))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l / (om / l)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -2.25e+91) {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l / (Om / l)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -2.25e+91:
		tmp = math.sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l / (Om / l)))))))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -2.25e+91)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -2.25e+91)
		tmp = sqrt((2.0 * (U * (n * (t + (((l * l) / Om) * -2.0))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l / (Om / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -2.25e+91], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -2.25e91

   1. Initial program 60.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in U around 0 47.5%

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

   5. Taylor expanded in n around 0 41.9%

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

      1. associate-*r* 56.2%

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

      2. unpow2 56.2%

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

   7. Simplified 56.2%

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

   if -2.25e91 < U

   1. Initial program 48.3%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in n around 0 45.8%

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

      1. *-commutative 45.8%

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

      2. unpow2 45.8%

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

      3. associate-/l* 49.7%

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

   6. Simplified 49.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.25 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]

Alternative 17: 38.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{+106}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \ell\right) \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)}}{Om}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.95e+106)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (* (* n l) (/ (sqrt (* 2.0 (* U (- U* U)))) Om))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.95e+106) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = (n * l) * (sqrt((2.0 * (U * (U_42_ - U)))) / Om);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.95d+106) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = (n * l) * (sqrt((2.0d0 * (u * (u_42 - u)))) / om)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.95e+106) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = (n * l) * (Math.sqrt((2.0 * (U * (U_42_ - U)))) / Om);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.95e+106:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = (n * l) * (math.sqrt((2.0 * (U * (U_42_ - U)))) / Om)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.95e+106)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = Float64(Float64(n * l) * Float64(sqrt(Float64(2.0 * Float64(U * Float64(U_42_ - U)))) / Om));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.95e+106)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = (n * l) * (sqrt((2.0 * (U * (U_42_ - U)))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.95e+106], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(n * l), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.94999999999999984e106

   1. Initial program 54.8%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in t around inf 39.0%

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

      1. pow1/2 40.9%

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

   6. Applied egg-rr 40.9%

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

   if 1.94999999999999984e106 < l

   1. Initial program 20.7%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in n around inf 19.5%

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

      1. associate-/l* 19.5%

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

      2. *-commutative 19.5%

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

   6. Simplified 19.5%

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

      1. associate-*l/ 19.5%

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

   8. Applied egg-rr 19.5%

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

      1. associate-/r/ 19.5%

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

      2. sqrt-unprod 19.5%

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

   10. Applied egg-rr 19.5%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{+106}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \ell\right) \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left(U* - U\right)\right)}}{Om}\\ \end{array} \]

Alternative 18: 38.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+108}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.25e+108)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (/ (sqrt (* (- U* U) (* 2.0 U))) (/ Om (* n l)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.25e+108) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.25d+108) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = sqrt(((u_42 - u) * (2.0d0 * u))) / (om / (n * l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.25e+108) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = Math.sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.25e+108:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = math.sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.25e+108)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(Float64(U_42_ - U) * Float64(2.0 * U))) / Float64(Om / Float64(n * l)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.25e+108)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = sqrt(((U_42_ - U) * (2.0 * U))) / (Om / (n * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.25e+108], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(N[(U$42$ - U), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.24999999999999998e108

   1. Initial program 54.8%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in t around inf 39.0%

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

      1. pow1/2 40.9%

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

   6. Applied egg-rr 40.9%

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

   if 1.24999999999999998e108 < l

   1. Initial program 20.7%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in n around inf 19.5%

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

      1. associate-/l* 19.5%

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

      2. *-commutative 19.5%

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

   6. Simplified 19.5%

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

      1. associate-*l/ 19.5%

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

   8. Applied egg-rr 19.5%

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

      1. div-inv 19.5%

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

      2. sqrt-unprod 19.5%

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

      3. associate-/r* 21.8%

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

   10. Applied egg-rr 21.8%

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

       1. associate-*r/ 21.8%

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

       2. *-rgt-identity 21.8%

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

       3. associate-*r* 21.8%

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

       4. associate-/l/ 19.5%

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

       5. *-commutative 19.5%

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

   12. Simplified 19.5%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+108}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* - U\right) \cdot \left(2 \cdot U\right)}}{\frac{Om}{n \cdot \ell}}\\ \end{array} \]

Alternative 19: 37.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq -2.1 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -2.1e+109)
   (sqrt (* (* (* 2.0 n) U) t))
   (pow (* (* 2.0 n) (* U t)) 0.5)))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -2.1e+109) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-2.1d+109)) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -2.1e+109) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -2.1e+109:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -2.1e+109)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -2.1e+109)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -2.1e+109], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -2.1000000000000001e109

   1. Initial program 62.4%

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

   2. Taylor expanded in t around inf 55.6%

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

   if -2.1000000000000001e109 < U

   1. Initial program 48.3%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around inf 34.1%

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

      1. pow1/2 35.9%

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

   6. Applied egg-rr 35.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.1 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 20: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -1e+108) (sqrt (* (* (* 2.0 n) U) t)) (sqrt (* (* 2.0 n) (* U t)))))

C

l = abs(l);
double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1e+108) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-1d+108)) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1e+108) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Python

l = abs(l)
def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -1e+108:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	return tmp

Julia

l = abs(l)
function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -1e+108)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -1e+108)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt(((2.0 * n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -1e+108], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1e108

   1. Initial program 62.4%

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

   2. Taylor expanded in t around inf 55.6%

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

   if -1e108 < U

   1. Initial program 48.3%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in t around inf 34.1%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 21: 35.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

3. Simplified 53.0%

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

4. Taylor expanded in t around inf 34.2%

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

5. Final simplification 34.2%

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

Reproduce

herbie shell --seed 2023238 
(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*))))))