Toniolo and Linder, Equation (13)

Percentage Accurate: 50.7% → 64.4%

Time: 26.5s

Alternatives: 22

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: 50.7% 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: 64.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (cbrt
          (sqrt
           (*
            U
            (+ t (* (/ l Om) (fma l -2.0 (* n (* (/ l Om) (- U* U)))))))))))
   (if (<= n -3e-140)
     (sqrt
      (*
       (* (* n 2.0) U)
       (+ t (* (fma l -2.0 (/ n (/ Om (* l U*)))) (/ l Om)))))
     (if (<= n -7.2e-274)
       (sqrt
        (-
         (* 2.0 (* n (* U t)))
         (*
          2.0
          (/
           (* (* n (* U l)) (- (/ (* n (* l (- U U*))) Om) (* l -2.0)))
           Om))))
       (if (<= n 1.65e-306)
         (sqrt (* 2.0 (* U (* n (+ t (/ (* -2.0 (* l l)) Om))))))
         (* (sqrt (* n 2.0)) (* t_1 (* t_1 t_1))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = cbrt(sqrt((U * (t + ((l / Om) * fma(l, -2.0, (n * ((l / Om) * (U_42_ - U)))))))));
	double tmp;
	if (n <= -3e-140) {
		tmp = sqrt((((n * 2.0) * U) * (t + (fma(l, -2.0, (n / (Om / (l * U_42_)))) * (l / Om)))));
	} else if (n <= -7.2e-274) {
		tmp = sqrt(((2.0 * (n * (U * t))) - (2.0 * (((n * (U * l)) * (((n * (l * (U - U_42_))) / Om) - (l * -2.0))) / Om))));
	} else if (n <= 1.65e-306) {
		tmp = sqrt((2.0 * (U * (n * (t + ((-2.0 * (l * l)) / Om))))));
	} else {
		tmp = sqrt((n * 2.0)) * (t_1 * (t_1 * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = cbrt(sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))))))))
	tmp = 0.0
	if (n <= -3e-140)
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * U) * Float64(t + Float64(fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_)))) * Float64(l / Om)))));
	elseif (n <= -7.2e-274)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) - Float64(2.0 * Float64(Float64(Float64(n * Float64(U * l)) * Float64(Float64(Float64(n * Float64(l * Float64(U - U_42_))) / Om) - Float64(l * -2.0))) / Om))));
	elseif (n <= 1.65e-306)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(-2.0 * Float64(l * l)) / Om))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * Float64(t_1 * Float64(t_1 * t_1)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[Sqrt[N[(U * 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], 1/3], $MachinePrecision]}, If[LessEqual[n, -3e-140], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -7.2e-274], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[(l * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.65e-306], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(-2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.00000000000000018e-140

   1. Initial program 52.7%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in U* around inf 53.8%

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

   5. Taylor expanded in U around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. *-commutative 53.8%

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

      3. fma-udef 53.8%

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

      4. associate-*l/ 53.8%

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

      5. *-commutative 53.8%

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

      6. associate-*r* 55.0%

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

      7. *-commutative 55.0%

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

      8. associate-/l* 57.5%

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

   7. Simplified 57.5%

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

   if -3.00000000000000018e-140 < n < -7.19999999999999965e-274

   1. Initial program 46.1%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in t around inf 72.0%

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

   if -7.19999999999999965e-274 < n < 1.65e-306

   1. Initial program 61.1%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in U* around inf 33.5%

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

   5. Taylor expanded in n around 0 33.5%

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

      1. associate-*r* 71.8%

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

      2. associate-*r/ 71.8%

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

      3. unpow2 71.8%

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

   7. Simplified 71.8%

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

   if 1.65e-306 < n

   1. Initial program 49.0%

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

   3. Simplified 57.1%

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

      1. sqrt-prod 65.0%

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

   5. Applied egg-rr 65.0%

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

      1. *-commutative 65.0%

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

      2. *-commutative 65.0%

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

   7. Simplified 65.0%

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

      1. add-cube-cbrt 64.0%

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

   9. Applied egg-rr 73.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \left(\sqrt[3]{\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}}\right)\right)\\ \end{array} \]

Alternative 2: 63.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot e^{\log \left(\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -7.5e-146)
   (sqrt
    (* (* (* n 2.0) U) (+ t (* (fma l -2.0 (/ n (/ Om (* l U*)))) (/ l Om)))))
   (if (<= n -5.6e-272)
     (sqrt
      (-
       (* 2.0 (* n (* U t)))
       (*
        2.0
        (/ (* (* n (* U l)) (- (/ (* n (* l (- U U*))) Om) (* l -2.0))) Om))))
     (if (<= n 1.65e-306)
       (sqrt (* 2.0 (* U (* n (+ t (/ (* -2.0 (* l l)) Om))))))
       (*
        (sqrt (* n 2.0))
        (exp
         (log
          (sqrt
           (*
            U
            (+
             t
             (* (/ l Om) (fma l -2.0 (* n (* (/ l Om) (- U* U)))))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -7.5e-146) {
		tmp = sqrt((((n * 2.0) * U) * (t + (fma(l, -2.0, (n / (Om / (l * U_42_)))) * (l / Om)))));
	} else if (n <= -5.6e-272) {
		tmp = sqrt(((2.0 * (n * (U * t))) - (2.0 * (((n * (U * l)) * (((n * (l * (U - U_42_))) / Om) - (l * -2.0))) / Om))));
	} else if (n <= 1.65e-306) {
		tmp = sqrt((2.0 * (U * (n * (t + ((-2.0 * (l * l)) / Om))))));
	} else {
		tmp = sqrt((n * 2.0)) * exp(log(sqrt((U * (t + ((l / Om) * fma(l, -2.0, (n * ((l / Om) * (U_42_ - U))))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -7.5e-146)
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * U) * Float64(t + Float64(fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_)))) * Float64(l / Om)))));
	elseif (n <= -5.6e-272)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) - Float64(2.0 * Float64(Float64(Float64(n * Float64(U * l)) * Float64(Float64(Float64(n * Float64(l * Float64(U - U_42_))) / Om) - Float64(l * -2.0))) / Om))));
	elseif (n <= 1.65e-306)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(Float64(-2.0 * Float64(l * l)) / Om))))));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * exp(log(sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(n * Float64(Float64(l / Om) * Float64(U_42_ - U)))))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -7.5e-146], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -5.6e-272], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[(l * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.65e-306], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t + N[(N[(-2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[Log[N[Sqrt[N[(U * 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]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -7.49999999999999981e-146

   1. Initial program 52.7%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in U* around inf 53.8%

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

   5. Taylor expanded in U around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. *-commutative 53.8%

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

      3. fma-udef 53.8%

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

      4. associate-*l/ 53.8%

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

      5. *-commutative 53.8%

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

      6. associate-*r* 55.0%

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

      7. *-commutative 55.0%

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

      8. associate-/l* 57.5%

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

   7. Simplified 57.5%

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

   if -7.49999999999999981e-146 < n < -5.59999999999999987e-272

   1. Initial program 46.1%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in t around inf 72.0%

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

   if -5.59999999999999987e-272 < n < 1.65e-306

   1. Initial program 61.1%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in U* around inf 33.5%

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

   5. Taylor expanded in n around 0 33.5%

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

      1. associate-*r* 71.8%

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

      2. associate-*r/ 71.8%

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

      3. unpow2 71.8%

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

   7. Simplified 71.8%

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

   if 1.65e-306 < n

   1. Initial program 49.0%

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

   3. Simplified 57.1%

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

      1. sqrt-prod 65.0%

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

   5. Applied egg-rr 65.0%

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

      1. *-commutative 65.0%

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

      2. *-commutative 65.0%

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

   7. Simplified 65.0%

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

      1. add-exp-log 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 71.4%

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

   9. Applied egg-rr 71.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot e^{\log \left(\sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\right)}\\ \end{array} \]

Alternative 3: 65.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot 2\right) \cdot U\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(U* - U\right) \cdot \left(n \cdot t_2\right)\\ t_4 := t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_3\right)\\ \mathbf{if}\;t_4 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot t_2\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n 2.0) U))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (- U* U) (* n t_2)))
        (t_4 (* t_1 (+ (- t (* 2.0 (/ (* l l) Om))) t_3))))
   (if (<= t_4 5e-324)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (- (* n (* (- U* U) t_2)) (* 2.0 (/ l (/ Om l))))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_1 (+ (- t (* 2.0 (* l (/ l Om)))) t_3)))
       (sqrt
        (*
         2.0
         (/ (* n (* l (* U (+ (* l -2.0) (/ (* n (* l U*)) Om))))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * 2.0) * U;
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (U_42_ - U) * (n * t_2);
	double t_4 = t_1 * ((t - (2.0 * ((l * l) / Om))) + t_3);
	double tmp;
	if (t_4 <= 5e-324) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n * ((U_42_ - U) * t_2)) - (2.0 * (l / (Om / l))))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_3)));
	} else {
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * 2.0) * U;
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = (U_42_ - U) * (n * t_2);
	double t_4 = t_1 * ((t - (2.0 * ((l * l) / Om))) + t_3);
	double tmp;
	if (t_4 <= 5e-324) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((n * ((U_42_ - U) * t_2)) - (2.0 * (l / (Om / l))))))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_3)));
	} else {
		tmp = Math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * 2.0) * U
	t_2 = math.pow((l / Om), 2.0)
	t_3 = (U_42_ - U) * (n * t_2)
	t_4 = t_1 * ((t - (2.0 * ((l * l) / Om))) + t_3)
	tmp = 0
	if t_4 <= 5e-324:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((n * ((U_42_ - U) * t_2)) - (2.0 * (l / (Om / l))))))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_3)))
	else:
		tmp = math.sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * 2.0) * U)
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(U_42_ - U) * Float64(n * t_2))
	t_4 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3))
	tmp = 0.0
	if (t_4 <= 5e-324)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n * Float64(Float64(U_42_ - U) * t_2)) - Float64(2.0 * Float64(l / Float64(Om / l))))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_3)));
	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

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * 2.0) * U;
	t_2 = (l / Om) ^ 2.0;
	t_3 = (U_42_ - U) * (n * t_2);
	t_4 = t_1 * ((t - (2.0 * ((l * l) / Om))) + t_3);
	tmp = 0.0;
	if (t_4 <= 5e-324)
		tmp = sqrt(((n * 2.0) * (U * (t + ((n * ((U_42_ - U) * t_2)) - (2.0 * (l / (Om / l))))))));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) + t_3)));
	else
		tmp = sqrt((2.0 * ((n * (l * (U * ((l * -2.0) + ((n * (l * U_42_)) / Om))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(U$42$ - U), $MachinePrecision] * N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-324], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $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 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 4.94066e-324

   1. Initial program 15.6%

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

      1. associate-*l* 37.6%

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

      2. sub-neg 37.6%

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

      3. associate-+l- 37.6%

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

      4. sub-neg 37.6%

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

      5. associate-/l* 41.9%

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

      6. remove-double-neg 41.9%

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

      7. associate-*l* 42.0%

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

   3. Simplified 42.0%

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

   if 4.94066e-324 < (*.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 67.1%

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

      1. associate-/l* 72.1%

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

      2. associate-/r/ 72.1%

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

   3. Applied egg-rr 72.1%

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

   if +inf.0 < (*.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 51.2%

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

   4. Taylor expanded in U* around inf 48.2%

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

   5. Taylor expanded in t around 0 58.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \left(\ell \cdot \left(U \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 4: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.3 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -2.3e+174)
   (sqrt
    (*
     (* n 2.0)
     (*
      U
      (+ t (- (* n (* (- U* U) (pow (/ l Om) 2.0))) (* 2.0 (/ l (/ Om l))))))))
   (sqrt
    (*
     (* (* n 2.0) U)
     (+ t (* (fma l -2.0 (/ n (/ Om (* l U*)))) (/ l Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -2.3e+174) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((n * ((U_42_ - U) * pow((l / Om), 2.0))) - (2.0 * (l / (Om / l))))))));
	} else {
		tmp = sqrt((((n * 2.0) * U) * (t + (fma(l, -2.0, (n / (Om / (l * U_42_)))) * (l / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -2.3e+174)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(n * Float64(Float64(U_42_ - U) * (Float64(l / Om) ^ 2.0))) - Float64(2.0 * Float64(l / Float64(Om / l))))))));
	else
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * U) * Float64(t + Float64(fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_)))) * Float64(l / Om)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -2.3e+174], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.2999999999999998e174

   1. Initial program 48.1%

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

      1. associate-*l* 57.8%

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

      2. sub-neg 57.8%

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

      3. associate-+l- 57.8%

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

      4. sub-neg 57.8%

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

      5. associate-/l* 77.1%

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

      6. remove-double-neg 77.1%

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

      7. associate-*l* 77.1%

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

   3. Simplified 77.1%

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

   if -2.2999999999999998e174 < Om

   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 55.0%

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

   4. Taylor expanded in U* around inf 56.8%

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

   5. Taylor expanded in U around 0 54.0%

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

      1. associate-*r* 54.1%

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

      2. *-commutative 54.1%

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

      3. fma-udef 54.1%

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

      4. associate-*l/ 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-*r* 58.2%

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

      7. *-commutative 58.2%

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

      8. associate-/l* 58.6%

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

   7. Simplified 58.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.3 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \end{array} \]

Alternative 5: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U* - U\right)\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{t_1}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- U* U))))
   (if (<= l -5.2e+26)
     (sqrt
      (* (* n 2.0) (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) t_1)))))))
     (if (<= l 1.1e+141)
       (sqrt
        (*
         (* (* n 2.0) U)
         (+ t (* (fma l -2.0 (/ n (/ Om (* l U*)))) (/ l Om)))))
       (if (<= l 1.12e+237)
         (sqrt
          (*
           -2.0
           (/ n (/ Om (* l (* (* U l) (+ 2.0 (* (- U U*) (/ n Om)))))))))
         (* (* l (sqrt 2.0)) (sqrt (/ (* n (* U (- (/ t_1 Om) 2.0))) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U_42_ - U);
	double tmp;
	if (l <= -5.2e+26) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * t_1)))))));
	} else if (l <= 1.1e+141) {
		tmp = sqrt((((n * 2.0) * U) * (t + (fma(l, -2.0, (n / (Om / (l * U_42_)))) * (l / Om)))));
	} else if (l <= 1.12e+237) {
		tmp = sqrt((-2.0 * (n / (Om / (l * ((U * l) * (2.0 + ((U - U_42_) * (n / Om)))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * ((t_1 / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U_42_ - U))
	tmp = 0.0
	if (l <= -5.2e+26)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * t_1)))))));
	elseif (l <= 1.1e+141)
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * U) * Float64(t + Float64(fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_)))) * Float64(l / Om)))));
	elseif (l <= 1.12e+237)
		tmp = sqrt(Float64(-2.0 * Float64(n / Float64(Om / Float64(l * Float64(Float64(U * l) * Float64(2.0 + Float64(Float64(U - U_42_) * Float64(n / Om)))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(t_1 / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.2e+26], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.1e+141], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.12e+237], N[Sqrt[N[(-2.0 * N[(n / N[(Om / N[(l * N[(N[(U * l), $MachinePrecision] * N[(2.0 + N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(t$95$1 / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.20000000000000004e26

   1. Initial program 26.8%

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

   3. Simplified 54.9%

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

   if -5.20000000000000004e26 < l < 1.1e141

   1. Initial program 62.0%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in U* around inf 60.5%

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

   5. Taylor expanded in U around 0 60.4%

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

      1. associate-*r* 60.5%

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

      2. *-commutative 60.5%

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

      3. fma-udef 60.5%

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

      4. associate-*l/ 60.5%

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

      5. *-commutative 60.5%

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

      6. associate-*r* 61.8%

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

      7. *-commutative 61.8%

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

      8. associate-/l* 63.9%

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

   7. Simplified 63.9%

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

   if 1.1e141 < l < 1.11999999999999997e237

   1. Initial program 7.6%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in l around -inf 33.6%

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

      1. associate-/l* 27.7%

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

      2. unpow2 27.7%

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

      3. *-commutative 27.7%

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

      4. mul-1-neg 27.7%

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

      5. associate-/l* 27.7%

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

   6. Simplified 27.7%

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

      1. pow1 27.7%

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

      2. associate-*l* 82.0%

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

      3. unsub-neg 82.0%

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

      4. associate-/r/ 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. unpow1 82.0%

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

      2. associate-*r* 82.0%

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

      3. *-commutative 82.0%

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

   10. Simplified 82.0%

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

   if 1.11999999999999997e237 < l

   1. Initial program 11.7%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in t around inf 33.4%

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

   5. Taylor expanded in l around inf 90.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 6: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.1e+106)
   (sqrt
    (-
     (* 2.0 (* n (* U t)))
     (*
      2.0
      (/ (* (* n (* U l)) (- (/ (* n (* l (- U U*))) Om) (* l -2.0))) Om))))
   (if (<= l 3.3e+134)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (fma -2.0 l (/ n (/ Om (* l U*))))) Om)))))
     (*
      (sqrt 2.0)
      (* l (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.1e+106) {
		tmp = sqrt(((2.0 * (n * (U * t))) - (2.0 * (((n * (U * l)) * (((n * (l * (U - U_42_))) / Om) - (l * -2.0))) / Om))));
	} else if (l <= 3.3e+134) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * fma(-2.0, l, (n / (Om / (l * U_42_))))) / Om)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U)))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.1e+106)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(U * t))) - Float64(2.0 * Float64(Float64(Float64(n * Float64(U * l)) * Float64(Float64(Float64(n * Float64(l * Float64(U - U_42_))) / Om) - Float64(l * -2.0))) / Om))));
	elseif (l <= 3.3e+134)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * fma(-2.0, l, Float64(n / Float64(Om / Float64(l * U_42_))))) / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.1e+106], N[Sqrt[N[(N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(n * N[(l * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.3e+134], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(-2.0 * l + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.09999999999999996e106

   1. Initial program 14.9%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around inf 47.4%

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

   if -1.09999999999999996e106 < l < 3.3e134

   1. Initial program 61.0%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in U around 0 59.9%

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

      1. *-commutative 59.9%

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

      2. fma-def 59.9%

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

      3. associate-/l* 62.9%

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

   6. Simplified 62.9%

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

   if 3.3e134 < l

   1. Initial program 12.5%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around 0 46.4%

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

   5. Taylor expanded in l around inf 65.4%

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

      1. associate-*l* 65.3%

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

      2. associate-/l* 64.1%

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

      3. *-commutative 64.1%

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

      4. sub-neg 64.1%

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

      5. associate-/l* 67.5%

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

      6. metadata-eval 67.5%

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

   7. Simplified 67.5%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) - 2 \cdot \frac{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\frac{n \cdot \left(\ell \cdot \left(U - U*\right)\right)}{Om} - \ell \cdot -2\right)}{Om}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\right)\\ \end{array} \]

Alternative 7: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{\frac{Om}{U \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.4e+111)
   (sqrt
    (* (* n 2.0) (/ (fma l -2.0 (* (/ l Om) (* n (- U* U)))) (/ Om (* U l)))))
   (if (<= l 6.4e+131)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (fma -2.0 l (/ n (/ Om (* l U*))))) Om)))))
     (*
      (sqrt 2.0)
      (* l (sqrt (/ n (/ Om (* U (+ -2.0 (/ n (/ Om (- U* U)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.4e+111) {
		tmp = sqrt(((n * 2.0) * (fma(l, -2.0, ((l / Om) * (n * (U_42_ - U)))) / (Om / (U * l)))));
	} else if (l <= 6.4e+131) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * fma(-2.0, l, (n / (Om / (l * U_42_))))) / Om)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt((n / (Om / (U * (-2.0 + (n / (Om / (U_42_ - U)))))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.4e+111)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))) / Float64(Om / Float64(U * l)))));
	elseif (l <= 6.4e+131)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l * fma(-2.0, l, Float64(n / Float64(Om / Float64(l * U_42_))))) / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(n / Float64(Om / Float64(U * Float64(-2.0 + Float64(n / Float64(Om / Float64(U_42_ - U))))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.4e+111], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.4e+131], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(-2.0 * l + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(n / N[(Om / N[(U * N[(-2.0 + N[(n / N[(Om / N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4e111

   1. Initial program 14.9%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in t around 0 32.8%

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

      1. associate-/l* 45.1%

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

      2. +-commutative 45.1%

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

      3. *-commutative 45.1%

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

      4. associate-*r* 45.5%

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

      5. *-commutative 45.5%

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

      6. associate-*r* 48.7%

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

      7. associate-*l/ 48.9%

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

      8. fma-udef 48.9%

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

      9. *-commutative 48.9%

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

   6. Simplified 48.9%

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

   if -1.4e111 < l < 6.4000000000000004e131

   1. Initial program 61.0%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in U around 0 59.9%

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

      1. *-commutative 59.9%

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

      2. fma-def 59.9%

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

      3. associate-/l* 62.9%

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

   6. Simplified 62.9%

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

   if 6.4000000000000004e131 < l

   1. Initial program 12.5%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around 0 46.4%

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

   5. Taylor expanded in l around inf 65.4%

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

      1. associate-*l* 65.3%

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

      2. associate-/l* 64.1%

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

      3. *-commutative 64.1%

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

      4. sub-neg 64.1%

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

      5. associate-/l* 67.5%

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

      6. metadata-eval 67.5%

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

   7. Simplified 67.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{\frac{Om}{U \cdot \ell}}}\\ \mathbf{elif}\;\ell \leq 6.4 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{n}{\frac{Om}{\ell \cdot U*}}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n}{\frac{Om}{U \cdot \left(-2 + \frac{n}{\frac{Om}{U* - U}}\right)}}}\right)\\ \end{array} \]

Alternative 8: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.55e+139)
   (sqrt
    (* (* (* n 2.0) U) (+ t (* (fma l -2.0 (/ n (/ Om (* l U*)))) (/ l Om)))))
   (if (<= l 1.12e+237)
     (sqrt
      (* -2.0 (/ n (/ Om (* l (* (* U l) (+ 2.0 (* (- U U*) (/ n Om)))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt (/ (* n (* U (- (/ (* n (- U* U)) Om) 2.0))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.55e+139) {
		tmp = sqrt((((n * 2.0) * U) * (t + (fma(l, -2.0, (n / (Om / (l * U_42_)))) * (l / Om)))));
	} else if (l <= 1.12e+237) {
		tmp = sqrt((-2.0 * (n / (Om / (l * ((U * l) * (2.0 + ((U - U_42_) * (n / Om)))))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt(((n * (U * (((n * (U_42_ - U)) / Om) - 2.0))) / Om));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.55e+139)
		tmp = sqrt(Float64(Float64(Float64(n * 2.0) * U) * Float64(t + Float64(fma(l, -2.0, Float64(n / Float64(Om / Float64(l * U_42_)))) * Float64(l / Om)))));
	elseif (l <= 1.12e+237)
		tmp = sqrt(Float64(-2.0 * Float64(n / Float64(Om / Float64(l * Float64(Float64(U * l) * Float64(2.0 + Float64(Float64(U - U_42_) * Float64(n / Om)))))))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Om) - 2.0))) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.55e+139], N[Sqrt[N[(N[(N[(n * 2.0), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(l * -2.0 + N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.12e+237], N[Sqrt[N[(-2.0 * N[(n / N[(Om / N[(l * N[(N[(U * l), $MachinePrecision] * N[(2.0 + N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * N[(U * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.55e139

   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 55.9%

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

   4. Taylor expanded in U* around inf 57.6%

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

   5. Taylor expanded in U around 0 55.4%

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

      1. associate-*r* 55.5%

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

      2. *-commutative 55.5%

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

      3. fma-udef 55.5%

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

      4. associate-*l/ 57.6%

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

      5. *-commutative 57.6%

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

      6. associate-*r* 57.3%

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

      7. *-commutative 57.3%

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

      8. associate-/l* 60.3%

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

   7. Simplified 60.3%

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

   if 1.55e139 < l < 1.11999999999999997e237

   1. Initial program 7.6%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in l around -inf 33.6%

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

      1. associate-/l* 27.7%

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

      2. unpow2 27.7%

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

      3. *-commutative 27.7%

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

      4. mul-1-neg 27.7%

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

      5. associate-/l* 27.7%

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

   6. Simplified 27.7%

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

      1. pow1 27.7%

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

      2. associate-*l* 82.0%

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

      3. unsub-neg 82.0%

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

      4. associate-/r/ 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. unpow1 82.0%

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

      2. associate-*r* 82.0%

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

      3. *-commutative 82.0%

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

   10. Simplified 82.0%

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

   if 1.11999999999999997e237 < l

   1. Initial program 11.7%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in t around inf 33.4%

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

   5. Taylor expanded in l around inf 90.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \mathsf{fma}\left(\ell, -2, \frac{n}{\frac{Om}{\ell \cdot U*}}\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)}{Om}}\\ \end{array} \]

Alternative 9: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -2.5e+163)
   (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt
    (*
     (* n 2.0)
     (* U (+ t (* (/ l Om) (fma l -2.0 (/ (* n (* l U*)) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -2.5e+163) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l / Om) * fma(l, -2.0, ((n * (l * U_42_)) / Om)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -2.5e+163)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(n * Float64(l * U_42_)) / Om)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -2.5e+163], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * 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]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.5e163

   1. Initial program 50.7%

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

      1. associate-*l* 57.3%

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

      2. sub-neg 57.3%

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

      3. associate-+l- 57.3%

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

      4. sub-neg 57.3%

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

      5. associate-/l* 74.8%

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

      6. remove-double-neg 74.8%

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

      7. associate-*l* 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in Om around inf 57.3%

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

      1. unpow2 57.3%

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

      2. associate-*r/ 70.3%

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

   6. Simplified 70.3%

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

   if -2.5e163 < Om

   1. Initial program 50.2%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in U* around inf 57.3%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 10: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+18} \lor \neg \left(n \leq 2.9 \cdot 10^{-50}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}{U*}}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -2.3e+18) (not (<= n 2.9e-50)))
   (sqrt (* (* n 2.0) (+ (* U t) (/ (/ n (/ (/ (/ (/ Om l) l) U) U*)) Om))))
   (sqrt
    (*
     2.0
     (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ n (/ Om (* U l)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -2.3e+18) || !(n <= 2.9e-50)) {
		tmp = sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((n <= (-2.3d+18)) .or. (.not. (n <= 2.9d-50))) then
        tmp = sqrt(((n * 2.0d0) * ((u * t) + ((n / ((((om / l) / l) / u) / u_42)) / om))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - (n / (om / (u * l)))))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -2.3e+18) || !(n <= 2.9e-50)) {
		tmp = Math.sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -2.3e+18) or not (n <= 2.9e-50):
		tmp = math.sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -2.3e+18) || !(n <= 2.9e-50))
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) + Float64(Float64(n / Float64(Float64(Float64(Float64(Om / l) / l) / U) / U_42_)) / Om))));
	else
		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)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -2.3e+18) || ~((n <= 2.9e-50)))
		tmp = sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -2.3e+18], N[Not[LessEqual[n, 2.9e-50]], $MachinePrecision]], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(n / N[(N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] / U), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if n < -2.3e18 or 2.90000000000000008e-50 < n

   1. Initial program 47.3%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in t around 0 46.6%

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

   5. Taylor expanded in U* around inf 42.7%

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

      1. associate-/l* 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*r* 47.9%

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

      4. associate-/r* 47.1%

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

      5. associate-/r* 48.7%

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

      6. unpow2 48.7%

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

      7. associate-/r* 52.0%

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

   7. Simplified 52.0%

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

   if -2.3e18 < n < 2.90000000000000008e-50

   1. Initial program 52.9%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in U* around 0 49.2%

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

      1. associate-*r* 53.0%

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

      2. +-commutative 53.0%

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

   6. Simplified 59.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+18} \lor \neg \left(n \leq 2.9 \cdot 10^{-50}\right):\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}{U*}}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \]

Alternative 11: 57.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.35 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.2e+77)
   (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (if (<= Om 2.35e+79)
     (sqrt
      (*
       (* n 2.0)
       (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
     (sqrt
      (*
       2.0
       (* U (* n (+ t (/ l (/ Om (- (* l -2.0) (/ n (/ Om (* U l))))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.2e+77) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else if (Om <= 2.35e+79) {
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (om <= (-1.2d+77)) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else if (om <= 2.35d+79) then
        tmp = sqrt(((n * 2.0d0) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t + (l / (om / ((l * (-2.0d0)) - (n / (om / (u * l)))))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.2e+77) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else if (Om <= 2.35e+79) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1.2e+77:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))))
	elif Om <= 2.35e+79:
		tmp = math.sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.2e+77)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	elseif (Om <= 2.35e+79)
		tmp = sqrt(Float64(Float64(n * 2.0) * 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(2.0 * Float64(U * Float64(n * Float64(t + Float64(l / Float64(Om / Float64(Float64(l * -2.0) - Float64(n / Float64(Om / Float64(U * l)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1.2e+77)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	elseif (Om <= 2.35e+79)
		tmp = sqrt(((n * 2.0) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t + (l / (Om / ((l * -2.0) - (n / (Om / (U * l)))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -1.2e+77], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.35e+79], N[Sqrt[N[(N[(n * 2.0), $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[(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]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.1999999999999999e77

   1. Initial program 49.0%

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

      1. associate-*l* 52.7%

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

      2. sub-neg 52.7%

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

      3. associate-+l- 52.7%

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

      4. sub-neg 52.7%

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

      5. associate-/l* 66.7%

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

      6. remove-double-neg 66.7%

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

      7. associate-*l* 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in Om around inf 52.8%

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

      1. unpow2 52.8%

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

      2. associate-*r/ 62.6%

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

   6. Simplified 62.6%

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

   if -1.1999999999999999e77 < Om < 2.35000000000000011e79

   1. Initial program 49.8%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in U around 0 59.8%

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

   if 2.35000000000000011e79 < Om

   1. Initial program 53.2%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in U* around 0 43.9%

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

      1. associate-*r* 50.0%

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

      2. +-commutative 50.0%

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

   6. Simplified 57.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 2.35 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell}{\frac{Om}{\ell \cdot -2 - \frac{n}{\frac{Om}{U \cdot \ell}}}}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 50.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.75 \cdot 10^{+114}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}{U*}}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -1.75e+114)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (if (<= U 1.6e-130)
     (sqrt (* (* n 2.0) (+ (* U t) (/ (/ n (/ (/ (/ (/ Om l) l) U) U*)) Om))))
     (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1.75e+114) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (U <= 1.6e-130) {
		tmp = sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (u <= (-1.75d+114)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (u <= 1.6d-130) then
        tmp = sqrt(((n * 2.0d0) * ((u * t) + ((n / ((((om / l) / l) / u) / u_42)) / om))))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * (l * (l / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1.75e+114) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (U <= 1.6e-130) {
		tmp = Math.sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -1.75e+114:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif U <= 1.6e-130:
		tmp = math.sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -1.75e+114)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (U <= 1.6e-130)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(U * t) + Float64(Float64(n / Float64(Float64(Float64(Float64(Om / l) / l) / U) / U_42_)) / Om))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -1.75e+114)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (U <= 1.6e-130)
		tmp = sqrt(((n * 2.0) * ((U * t) + ((n / ((((Om / l) / l) / U) / U_42_)) / Om))));
	else
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -1.75e+114], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[U, 1.6e-130], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(n / N[(N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] / U), $MachinePrecision] / U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -1.75e114

   1. Initial program 49.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in t around inf 34.9%

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

      1. pow1/2 38.6%

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

      2. associate-*l* 38.6%

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

      3. associate-*r* 54.6%

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

   6. Applied egg-rr 54.6%

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

   if -1.75e114 < U < 1.6e-130

   1. Initial program 43.0%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in t around 0 53.0%

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

   5. Taylor expanded in U* around inf 45.8%

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

      1. associate-/l* 45.9%

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

      2. *-commutative 45.9%

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

      3. associate-*r* 46.7%

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

      4. associate-/r* 46.7%

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

      5. associate-/r* 48.0%

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

      6. unpow2 48.0%

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

      7. associate-/r* 50.7%

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

   7. Simplified 50.7%

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

   if 1.6e-130 < U

   1. Initial program 64.5%

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

      1. associate-*l* 59.6%

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

      2. sub-neg 59.6%

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

      3. associate-+l- 59.6%

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

      4. sub-neg 59.6%

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

      5. associate-/l* 67.0%

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

      6. remove-double-neg 67.0%

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

      7. associate-*l* 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in Om around inf 56.0%

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

      1. unpow2 56.0%

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

      2. associate-*r/ 63.3%

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

   6. Simplified 63.3%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.75 \cdot 10^{+114}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot t + \frac{\frac{n}{\frac{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}{U*}}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13: 50.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3e+144)
   (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt
    (* -2.0 (* (* l (/ n Om)) (* (* U l) (+ 2.0 (/ (* n (- U U*)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e+144) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * ((l * (n / Om)) * ((U * l) * (2.0 + ((n * (U - U_42_)) / Om))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 3d+144) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * ((l * (n / om)) * ((u * l) * (2.0d0 + ((n * (u - u_42)) / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e+144) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((l * (n / Om)) * ((U * l) * (2.0 + ((n * (U - U_42_)) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3e+144:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * ((l * (n / Om)) * ((U * l) * (2.0 + ((n * (U - U_42_)) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3e+144)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(l * Float64(n / Om)) * Float64(Float64(U * l) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3e+144)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * ((l * (n / Om)) * ((U * l) * (2.0 + ((n * (U - U_42_)) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3e+144], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(U * l), $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9999999999999999e144

   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. Step-by-step derivation

      1. associate-*l* 54.6%

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

      2. sub-neg 54.6%

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

      3. associate-+l- 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-/l* 58.0%

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

      6. remove-double-neg 58.0%

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

      7. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in Om around inf 48.0%

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

      1. unpow2 48.0%

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

      2. associate-*r/ 50.9%

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

   6. Simplified 50.9%

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

   if 2.9999999999999999e144 < l

   1. Initial program 9.3%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in l around -inf 35.3%

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

      1. associate-/l* 31.5%

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

      2. unpow2 31.5%

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

      3. *-commutative 31.5%

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

      4. mul-1-neg 31.5%

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

      5. associate-/l* 31.5%

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

   6. Simplified 31.5%

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

      1. *-un-lft-identity 31.5%

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

      2. associate-/r/ 34.7%

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

      3. associate-*l* 54.5%

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

      4. unsub-neg 54.5%

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

      5. associate-/r/ 54.5%

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

   8. Applied egg-rr 54.5%

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

      1. *-lft-identity 54.5%

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

      2. associate-*r* 50.6%

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

      3. associate-*r* 50.6%

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

      4. associate-*l/ 46.8%

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

   10. Simplified 46.8%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 14: 49.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e+151)
   (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt
    (* -2.0 (/ n (/ Om (* l (* (* U l) (+ 2.0 (* (- U U*) (/ n Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.5e+151) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * (n / (Om / (l * ((U * l) * (2.0 + ((U - U_42_) * (n / Om)))))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 4.5d+151) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * (n / (om / (l * ((u * l) * (2.0d0 + ((u - u_42) * (n / om)))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.5e+151) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (n / (Om / (l * ((U * l) * (2.0 + ((U - U_42_) * (n / Om)))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.5e+151:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * (n / (Om / (l * ((U * l) * (2.0 + ((U - U_42_) * (n / Om)))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.5e+151)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(n / Float64(Om / Float64(l * Float64(Float64(U * l) * Float64(2.0 + Float64(Float64(U - U_42_) * Float64(n / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.5e+151)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * (n / (Om / (l * ((U * l) * (2.0 + ((U - U_42_) * (n / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.5e+151], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(n / N[(Om / N[(l * N[(N[(U * l), $MachinePrecision] * N[(2.0 + N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.4999999999999999e151

   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. Step-by-step derivation

      1. associate-*l* 54.6%

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

      2. sub-neg 54.6%

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

      3. associate-+l- 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-/l* 58.0%

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

      6. remove-double-neg 58.0%

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

      7. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in Om around inf 48.0%

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

      1. unpow2 48.0%

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

      2. associate-*r/ 50.9%

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

   6. Simplified 50.9%

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

   if 4.4999999999999999e151 < l

   1. Initial program 9.3%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in l around -inf 35.3%

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

      1. associate-/l* 31.5%

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

      2. unpow2 31.5%

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

      3. *-commutative 31.5%

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

      4. mul-1-neg 31.5%

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

      5. associate-/l* 31.5%

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

   6. Simplified 31.5%

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

      1. pow1 31.5%

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

      2. associate-*l* 66.3%

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

      3. unsub-neg 66.3%

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

      4. associate-/r/ 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. unpow1 66.3%

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

      2. associate-*r* 66.3%

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

      3. *-commutative 66.3%

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

   10. Simplified 66.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\ell \cdot \left(\left(U \cdot \ell\right) \cdot \left(2 + \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)}}}\\ \end{array} \]

Alternative 15: 48.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + U \cdot \frac{n}{Om}\right)\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e+148)
   (sqrt (* (* n 2.0) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (pow (* -2.0 (* (/ n Om) (* l (* l (* U (+ 2.0 (* U (/ n Om)))))))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e+148) {
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + (U * (n / Om)))))))), 0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 1d+148) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = ((-2.0d0) * ((n / om) * (l * (l * (u * (2.0d0 + (u * (n / om)))))))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e+148) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + (U * (n / Om)))))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1e+148:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow((-2.0 * ((n / Om) * (l * (l * (U * (2.0 + (U * (n / Om)))))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e+148)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(-2.0 * Float64(Float64(n / Om) * Float64(l * Float64(l * Float64(U * Float64(2.0 + Float64(U * Float64(n / Om)))))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1e+148)
		tmp = sqrt(((n * 2.0) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = (-2.0 * ((n / Om) * (l * (l * (U * (2.0 + (U * (n / Om)))))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e+148], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-2.0 * N[(N[(n / Om), $MachinePrecision] * N[(l * N[(l * N[(U * N[(2.0 + N[(U * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1e148

   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. Step-by-step derivation

      1. associate-*l* 54.6%

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

      2. sub-neg 54.6%

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

      3. associate-+l- 54.6%

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

      4. sub-neg 54.6%

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

      5. associate-/l* 58.0%

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

      6. remove-double-neg 58.0%

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

      7. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in Om around inf 48.0%

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

      1. unpow2 48.0%

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

      2. associate-*r/ 50.9%

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

   6. Simplified 50.9%

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

   if 1e148 < l

   1. Initial program 9.3%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in l around -inf 35.3%

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

      1. associate-/l* 31.5%

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

      2. unpow2 31.5%

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

      3. *-commutative 31.5%

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

      4. mul-1-neg 31.5%

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

      5. associate-/l* 31.5%

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

   6. Simplified 31.5%

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

   7. Taylor expanded in U* around 0 2.3%

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

      1. associate-/r* 2.3%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{2 - -1 \cdot \frac{n \cdot U}{Om}}}{{\ell}^{2} \cdot U}}}} \]

      2. sub-neg 2.3%

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

      3. mul-1-neg 2.3%

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

      4. remove-double-neg 2.3%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{2 + \color{blue}{\frac{n \cdot U}{Om}}}}{{\ell}^{2} \cdot U}}} \]

      5. associate-/l* 2.3%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{2 + \color{blue}{\frac{n}{\frac{Om}{U}}}}}{{\ell}^{2} \cdot U}}} \]

      6. *-commutative 2.3%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{2 + \frac{n}{\frac{Om}{U}}}}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

      7. unpow2 2.3%

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

   9. Simplified 2.3%

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

   10. Taylor expanded in l around 0 6.2%

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

       1. associate-/l* 2.3%

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

       2. unpow2 2.3%

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

       3. *-commutative 2.3%

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

       4. associate-*l/ 2.3%

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

       5. *-commutative 2.3%

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

   12. Simplified 2.3%

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

       1. pow1/2 31.5%

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

       2. associate-/r/ 34.7%

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

       3. associate-*l* 46.6%

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

   14. Applied egg-rr 46.6%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \left(\frac{n}{Om} \cdot \left(\ell \cdot \left(\ell \cdot \left(U \cdot \left(2 + U \cdot \frac{n}{Om}\right)\right)\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 16: 40.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-199} \lor \neg \left(t \leq 5 \cdot 10^{-167}\right):\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot 2}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= t -1.9e-199) (not (<= t 5e-167)))
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* -2.0 (/ (* n 2.0) (/ (/ (/ Om l) l) U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((t <= -1.9e-199) || !(t <= 5e-167)) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((-2.0 * ((n * 2.0) / (((Om / l) / l) / U))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t <= (-1.9d-199)) .or. (.not. (t <= 5d-167))) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt(((-2.0d0) * ((n * 2.0d0) / (((om / l) / l) / u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((t <= -1.9e-199) || !(t <= 5e-167)) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * ((n * 2.0) / (((Om / l) / l) / U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (t <= -1.9e-199) or not (t <= 5e-167):
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * ((n * 2.0) / (((Om / l) / l) / U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((t <= -1.9e-199) || !(t <= 5e-167))
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * 2.0) / Float64(Float64(Float64(Om / l) / l) / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((t <= -1.9e-199) || ~((t <= 5e-167)))
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * ((n * 2.0) / (((Om / l) / l) / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[t, -1.9e-199], N[Not[LessEqual[t, 5e-167]], $MachinePrecision]], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(n * 2.0), $MachinePrecision] / N[(N[(N[(Om / l), $MachinePrecision] / l), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8999999999999999e-199 or 5.0000000000000002e-167 < t

   1. Initial program 52.8%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in t around inf 43.3%

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

      1. pow1/2 44.4%

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

      2. associate-*l* 43.9%

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

      3. associate-*r* 47.2%

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

   6. Applied egg-rr 47.2%

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

   if -1.8999999999999999e-199 < t < 5.0000000000000002e-167

   1. Initial program 40.8%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in l around -inf 36.4%

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

      1. associate-/l* 38.2%

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

      2. unpow2 38.2%

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

      3. *-commutative 38.2%

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

      4. mul-1-neg 38.2%

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

      5. associate-/l* 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in U* around 0 27.9%

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

      1. associate-/r* 27.9%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\color{blue}{\frac{\frac{Om}{2 - -1 \cdot \frac{n \cdot U}{Om}}}{{\ell}^{2} \cdot U}}}} \]

      2. sub-neg 27.9%

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

      3. mul-1-neg 27.9%

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

      4. remove-double-neg 27.9%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{2 + \color{blue}{\frac{n \cdot U}{Om}}}}{{\ell}^{2} \cdot U}}} \]

      5. associate-/l* 27.9%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{2 + \color{blue}{\frac{n}{\frac{Om}{U}}}}}{{\ell}^{2} \cdot U}}} \]

      6. *-commutative 27.9%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{\frac{Om}{2 + \frac{n}{\frac{Om}{U}}}}{\color{blue}{U \cdot {\ell}^{2}}}}} \]

      7. unpow2 27.9%

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

   9. Simplified 27.9%

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

   10. Taylor expanded in l around 0 25.9%

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

       1. associate-/l* 27.9%

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

       2. unpow2 27.9%

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

       3. *-commutative 27.9%

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

       4. associate-*l/ 27.9%

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

       5. *-commutative 27.9%

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

   12. Simplified 27.9%

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

   13. Taylor expanded in n around 0 26.2%

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

       1. associate-/l* 28.1%

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

       2. associate-*r/ 28.1%

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

       3. associate-/r* 28.1%

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

       4. unpow2 28.1%

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

       5. associate-/r* 35.3%

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

   15. Simplified 35.3%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-199} \lor \neg \left(t \leq 5 \cdot 10^{-167}\right):\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n \cdot 2}{\frac{\frac{\frac{Om}{\ell}}{\ell}}{U}}}\\ \end{array} \]

Alternative 17: 45.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 * (u * (n * (t + (((-2.0d0) * (l * l)) / om))))))
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 * (U * (n * (t + ((-2.0 * (l * l)) / Om))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 55.5%

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

4. Taylor expanded in U* around inf 56.0%

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

5. Taylor expanded in n around 0 44.2%

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

   1. associate-*r* 45.2%

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

   2. associate-*r/ 45.2%

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

   3. unpow2 45.2%

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

7. Simplified 45.2%

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

8. Final simplification 45.2%

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

Alternative 18: 47.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. associate-*l* 50.2%

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

   2. sub-neg 50.2%

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

   3. associate-+l- 50.2%

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

   4. sub-neg 50.2%

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

   5. associate-/l* 55.9%

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

   6. remove-double-neg 55.9%

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

   7. associate-*l* 56.3%

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

3. Simplified 56.3%

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

4. Taylor expanded in Om around inf 44.3%

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

   1. unpow2 44.3%

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

   2. associate-*r/ 48.9%

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

6. Simplified 48.9%

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

7. Final simplification 48.9%

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

Alternative 19: 37.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -7 \cdot 10^{-309}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -7e-309)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* 2.0 (* t (* n U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -7e-309) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (u_42 <= (-7d-309)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -7e-309) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -7e-309:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -7e-309)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -7e-309)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -7e-309], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -6.9999999999999984e-309

   1. Initial program 41.8%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in l around 0 35.0%

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

      1. sqrt-unprod 35.2%

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

      2. associate-*r* 35.2%

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

   6. Applied egg-rr 35.2%

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

      1. pow1/2 37.1%

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

      2. associate-*r* 37.0%

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

      3. *-commutative 37.0%

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

   8. Applied egg-rr 37.0%

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

   if -6.9999999999999984e-309 < U*

   1. Initial program 57.5%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in l around 0 38.7%

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

      1. sqrt-unprod 38.9%

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

      2. associate-*r* 43.2%

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

   6. Applied egg-rr 43.2%

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

      1. pow1/2 44.0%

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

      2. associate-*r* 39.0%

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

      3. *-commutative 39.0%

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

   8. Applied egg-rr 39.0%

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

      1. unpow1/2 38.9%

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

      2. associate-*r* 45.2%

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

   10. Applied egg-rr 45.2%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -7 \cdot 10^{-309}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 20: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -3.3 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -3.3e-229)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* 2.0 (* t (* n U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -3.3e-229) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (u_42 <= (-3.3d-229)) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt((2.0d0 * (t * (n * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -3.3e-229) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -3.3e-229:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -3.3e-229)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -3.3e-229)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -3.3e-229], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -3.30000000000000021e-229

   1. Initial program 40.0%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in l around 0 32.6%

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

      1. sqrt-unprod 32.7%

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

      2. associate-*r* 34.2%

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

   6. Applied egg-rr 34.2%

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

   if -3.30000000000000021e-229 < U*

   1. Initial program 57.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in l around 0 40.3%

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

      1. sqrt-unprod 40.4%

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

      2. associate-*r* 43.4%

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

   6. Applied egg-rr 43.4%

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

      1. pow1/2 44.2%

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

      2. associate-*r* 40.5%

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

      3. *-commutative 40.5%

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

   8. Applied egg-rr 40.5%

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

      1. unpow1/2 40.4%

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

      2. associate-*r* 46.3%

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

   10. Applied egg-rr 46.3%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -3.3 \cdot 10^{-229}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 21: 38.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* U (* n t))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (U * (n * t))), 0.5);
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (2.0d0 * (u * (n * t))) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((2.0 * (U * (n * t))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (U * (n * t))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * (U * (n * t))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 50.3%

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

3. Simplified 55.5%

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

4. Taylor expanded in t around inf 37.6%

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

   1. pow1/2 38.5%

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

   2. associate-*l* 38.1%

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

   3. associate-*r* 40.9%

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

6. Applied egg-rr 40.9%

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

7. Final simplification 40.9%

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

Alternative 22: 36.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 * (t * (n * u))))
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 * (t * (n * U))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Simplified 55.5%

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

4. Taylor expanded in l around 0 37.0%

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

   1. sqrt-unprod 37.2%

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

   2. associate-*r* 39.5%

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

6. Applied egg-rr 39.5%

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

   1. pow1/2 40.9%

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

   2. associate-*r* 38.1%

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

   3. *-commutative 38.1%

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

8. Applied egg-rr 38.1%

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

   1. unpow1/2 37.2%

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

   2. associate-*r* 38.2%

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

10. Applied egg-rr 38.2%

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

11. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2023194 
(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*))))))