Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 66.1%

Time: 50.9s

Alternatives: 19

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: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_4 := t\_2 \cdot \left(t\_3 + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_4 \leq 10^{-321}:\\ \;\;\;\;\sqrt{U \cdot t\_1} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t\_3 + \left(\frac{\ell}{Om} \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))
        (t_2 (* (* 2.0 n) U))
        (t_3 (- t (* 2.0 (/ (* l l) Om))))
        (t_4 (* t_2 (+ t_3 (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_4 1e-321)
     (* (sqrt (* U t_1)) (sqrt n))
     (if (<= t_4 5e+293)
       (sqrt (* t_2 (+ t_3 (* (* (/ l Om) (/ n (/ Om l))) (- U* U)))))
       (if (<= t_4 INFINITY)
         (sqrt (* U (* n t_1)))
         (sqrt
          (/
           (* 2.0 (* U (* (* n l) (+ (* l -2.0) (* U* (/ (* n l) Om))))))
           Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)));
	double t_2 = (2.0 * n) * U;
	double t_3 = t - (2.0 * ((l * l) / Om));
	double t_4 = t_2 * (t_3 + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_4 <= 1e-321) {
		tmp = sqrt((U * t_1)) * sqrt(n);
	} else if (t_4 <= 5e+293) {
		tmp = sqrt((t_2 * (t_3 + (((l / Om) * (n / (Om / l))) * (U_42_ - U)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((U * (n * t_1)));
	} else {
		tmp = sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) + (U_42_ * ((n * l) / 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 = 2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)));
	double t_2 = (2.0 * n) * U;
	double t_3 = t - (2.0 * ((l * l) / Om));
	double t_4 = t_2 * (t_3 + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_4 <= 1e-321) {
		tmp = Math.sqrt((U * t_1)) * Math.sqrt(n);
	} else if (t_4 <= 5e+293) {
		tmp = Math.sqrt((t_2 * (t_3 + (((l / Om) * (n / (Om / l))) * (U_42_ - U)))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((U * (n * t_1)));
	} else {
		tmp = Math.sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) + (U_42_ * ((n * l) / Om)))))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))
	t_2 = (2.0 * n) * U
	t_3 = t - (2.0 * ((l * l) / Om))
	t_4 = t_2 * (t_3 + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))
	tmp = 0
	if t_4 <= 1e-321:
		tmp = math.sqrt((U * t_1)) * math.sqrt(n)
	elif t_4 <= 5e+293:
		tmp = math.sqrt((t_2 * (t_3 + (((l / Om) * (n / (Om / l))) * (U_42_ - U)))))
	elif t_4 <= math.inf:
		tmp = math.sqrt((U * (n * t_1)))
	else:
		tmp = math.sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) + (U_42_ * ((n * l) / Om)))))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l))))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_4 = Float64(t_2 * Float64(t_3 + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_4 <= 1e-321)
		tmp = Float64(sqrt(Float64(U * t_1)) * sqrt(n));
	elseif (t_4 <= 5e+293)
		tmp = sqrt(Float64(t_2 * Float64(t_3 + Float64(Float64(Float64(l / Om) * Float64(n / Float64(Om / l))) * Float64(U_42_ - U)))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(U * Float64(n * t_1)));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(n * l) * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om)))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)));
	t_2 = (2.0 * n) * U;
	t_3 = t - (2.0 * ((l * l) / Om));
	t_4 = t_2 * (t_3 + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)));
	tmp = 0.0;
	if (t_4 <= 1e-321)
		tmp = sqrt((U * t_1)) * sqrt(n);
	elseif (t_4 <= 5e+293)
		tmp = sqrt((t_2 * (t_3 + (((l / Om) * (n / (Om / l))) * (U_42_ - U)))));
	elseif (t_4 <= Inf)
		tmp = sqrt((U * (n * t_1)));
	else
		tmp = sqrt(((2.0 * (U * ((n * l) * ((l * -2.0) + (U_42_ * ((n * l) / Om)))))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(t + N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$3 + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-321], N[(N[Sqrt[N[(U * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+293], N[Sqrt[N[(t$95$2 * N[(t$95$3 + N[(N[(N[(l / Om), $MachinePrecision] * N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.98013e-322

   1. Initial program 14.3%

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

   3. Simplified 28.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U}\right), \color{blue}{\left(\sqrt{n}\right)}\right) \]

   6. Applied egg-rr 57.2%

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

   if 9.98013e-322 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000033e293

   1. Initial program 98.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \left(\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\left(U - U*\right), \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{\ell}{Om} \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(\frac{n \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\frac{n \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(n \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\frac{n}{\frac{Om}{\ell}}\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 98.9%

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

   if 5.00000000000000033e293 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 32.2%

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

   3. Simplified 42.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot n\right) \cdot U\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot n\right), U\right)\right) \]

   6. Applied egg-rr 46.9%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.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 21.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 25.4%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 26.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 26.1%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right), Om\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right), Om\right)\right) \]

       13. *-lowering-*.f64 66.3%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right), Om\right)\right) \]

   12. Simplified 66.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-321}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \frac{n}{\frac{Om}{\ell}}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(\frac{U \cdot \ell}{Om} \cdot \frac{n \cdot t\_1}{t} + n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(t + \frac{t\_1}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ (* l -2.0) (* U* (/ (* n l) Om)))))
   (if (<= n -6.2e-86)
     (sqrt
      (*
       n
       (* U (* 2.0 (+ t (/ (+ (* l -2.0) (* U* (* l (/ n Om)))) (/ Om l)))))))
     (if (<= n 9.2e-308)
       (sqrt (* t (* 2.0 (+ (* (/ (* U l) Om) (/ (* n t_1) t)) (* n U)))))
       (* (sqrt n) (sqrt (* 2.0 (* U (+ t (/ t_1 (/ Om l)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * -2.0) + (U_42_ * ((n * l) / Om));
	double tmp;
	if (n <= -6.2e-86) {
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else if (n <= 9.2e-308) {
		tmp = sqrt((t * (2.0 * ((((U * l) / Om) * ((n * t_1) / t)) + (n * U)))));
	} else {
		tmp = sqrt(n) * sqrt((2.0 * (U * (t + (t_1 / (Om / 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) :: t_1
    real(8) :: tmp
    t_1 = (l * (-2.0d0)) + (u_42 * ((n * l) / om))
    if (n <= (-6.2d-86)) then
        tmp = sqrt((n * (u * (2.0d0 * (t + (((l * (-2.0d0)) + (u_42 * (l * (n / om)))) / (om / l)))))))
    else if (n <= 9.2d-308) then
        tmp = sqrt((t * (2.0d0 * ((((u * l) / om) * ((n * t_1) / t)) + (n * u)))))
    else
        tmp = sqrt(n) * sqrt((2.0d0 * (u * (t + (t_1 / (om / 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 t_1 = (l * -2.0) + (U_42_ * ((n * l) / Om));
	double tmp;
	if (n <= -6.2e-86) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else if (n <= 9.2e-308) {
		tmp = Math.sqrt((t * (2.0 * ((((U * l) / Om) * ((n * t_1) / t)) + (n * U)))));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (U * (t + (t_1 / (Om / l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * -2.0) + (U_42_ * ((n * l) / Om))
	tmp = 0
	if n <= -6.2e-86:
		tmp = math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))))
	elif n <= 9.2e-308:
		tmp = math.sqrt((t * (2.0 * ((((U * l) / Om) * ((n * t_1) / t)) + (n * U)))))
	else:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (U * (t + (t_1 / (Om / l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om)))
	tmp = 0.0
	if (n <= -6.2e-86)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(l * Float64(n / Om)))) / Float64(Om / l)))))));
	elseif (n <= 9.2e-308)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(Float64(Float64(Float64(U * l) / Om) * Float64(Float64(n * t_1) / t)) + Float64(n * U)))));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * Float64(t + Float64(t_1 / Float64(Om / l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * -2.0) + (U_42_ * ((n * l) / Om));
	tmp = 0.0;
	if (n <= -6.2e-86)
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	elseif (n <= 9.2e-308)
		tmp = sqrt((t * (2.0 * ((((U * l) / Om) * ((n * t_1) / t)) + (n * U)))));
	else
		tmp = sqrt(n) * sqrt((2.0 * (U * (t + (t_1 / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-86], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 9.2e-308], N[Sqrt[N[(t * N[(2.0 * N[(N[(N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(n * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * N[(t + N[(t$95$1 / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.19999999999999977e-86

   1. Initial program 60.0%

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

   3. Simplified 64.1%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 69.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 69.3%

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

   if -6.19999999999999977e-86 < n < 9.1999999999999996e-308

   1. Initial program 44.3%

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

   3. Simplified 52.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 52.0%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 52.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 52.1%

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

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right) \]

       2. distribute-lft-out N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(U \cdot n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{\left(U \cdot \ell\right) \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       7. times-frac N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U \cdot \ell}{Om} \cdot \frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\left(\frac{U \cdot \ell}{Om}\right), \left(\frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \ell\right), Om\right), \left(\frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \ell\right), Om\right), \left(\frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \ell\right), Om\right), \mathsf{/.f64}\left(\left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right), t\right)\right)\right)\right)\right)\right) \]

   12. Simplified 63.9%

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

   if 9.1999999999999996e-308 < n

   1. Initial program 47.5%

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

   3. Simplified 55.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 57.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 57.8%

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

       1. pow1/2 N/A

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

       2. unpow-prod-down N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(2 \cdot \left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot U\right)}^{\frac{1}{2}}\right), \color{blue}{\left({n}^{\frac{1}{2}}\right)}\right) \]

   11. Applied egg-rr 69.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(\frac{U \cdot \ell}{Om} \cdot \frac{n \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{t} + n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(\frac{U \cdot \ell}{Om} \cdot \frac{n \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{t} + n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -8e-86)
   (sqrt
    (*
     n
     (* U (* 2.0 (+ t (/ (+ (* l -2.0) (* U* (* l (/ n Om)))) (/ Om l)))))))
   (if (<= n 1.35e-41)
     (sqrt
      (*
       t
       (*
        2.0
        (+
         (* (/ (* U l) Om) (/ (* n (+ (* l -2.0) (* U* (/ (* n l) Om)))) t))
         (* n U)))))
     (sqrt
      (*
       n
       (*
        U
        (*
         2.0
         (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -8e-86) {
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else if (n <= 1.35e-41) {
		tmp = sqrt((t * (2.0 * ((((U * l) / Om) * ((n * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / t)) + (n * U)))));
	} else {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / 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 <= (-8d-86)) then
        tmp = sqrt((n * (u * (2.0d0 * (t + (((l * (-2.0d0)) + (u_42 * (l * (n / om)))) / (om / l)))))))
    else if (n <= 1.35d-41) then
        tmp = sqrt((t * (2.0d0 * ((((u * l) / om) * ((n * ((l * (-2.0d0)) + (u_42 * ((n * l) / om)))) / t)) + (n * u)))))
    else
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / 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 <= -8e-86) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else if (n <= 1.35e-41) {
		tmp = Math.sqrt((t * (2.0 * ((((U * l) / Om) * ((n * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / t)) + (n * U)))));
	} else {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -8e-86:
		tmp = math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))))
	elif n <= 1.35e-41:
		tmp = math.sqrt((t * (2.0 * ((((U * l) / Om) * ((n * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / t)) + (n * U)))))
	else:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -8e-86)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(l * Float64(n / Om)))) / Float64(Om / l)))))));
	elseif (n <= 1.35e-41)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(Float64(Float64(Float64(U * l) / Om) * Float64(Float64(n * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om)))) / t)) + Float64(n * U)))));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -8e-86)
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	elseif (n <= 1.35e-41)
		tmp = sqrt((t * (2.0 * ((((U * l) / Om) * ((n * ((l * -2.0) + (U_42_ * ((n * l) / Om)))) / t)) + (n * U)))));
	else
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -8e-86], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.35e-41], N[Sqrt[N[(t * N[(2.0 * N[(N[(N[(N[(U * l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(n * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.00000000000000068e-86

   1. Initial program 60.0%

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

   3. Simplified 64.1%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 69.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 69.3%

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

   if -8.00000000000000068e-86 < n < 1.35e-41

   1. Initial program 44.6%

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

   3. Simplified 52.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 52.0%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 52.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 52.0%

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

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(t \cdot \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n\right) + 2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right) \]

       2. distribute-lft-out N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \left(U \cdot n + \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(U \cdot n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{\left(U \cdot \ell\right) \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om \cdot t}\right)\right)\right)\right)\right) \]

       7. times-frac N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(\frac{U \cdot \ell}{Om} \cdot \frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\left(\frac{U \cdot \ell}{Om}\right), \left(\frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \ell\right), Om\right), \left(\frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \ell\right), Om\right), \left(\frac{n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{t}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \ell\right), Om\right), \mathsf{/.f64}\left(\left(n \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right), t\right)\right)\right)\right)\right)\right) \]

   12. Simplified 66.9%

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

   if 1.35e-41 < n

   1. Initial program 49.3%

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

   3. Simplified 58.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 63.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(\frac{U \cdot \ell}{Om} \cdot \frac{n \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)}{t} + n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.44 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.6e-72)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= l 1.44e+47)
     (sqrt (* (* 2.0 U) (* n (- t (* 2.0 (/ (* l l) Om))))))
     (sqrt
      (/ (* (* (* U -2.0) (+ 2.0 (/ n (/ Om (- U U*))))) (* l (* n l))) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.6e-72) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 1.44e+47) {
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * 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 (l <= 3.6d-72) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 1.44d+47) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt(((((u * (-2.0d0)) * (2.0d0 + (n / (om / (u - u_42))))) * (l * (n * 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 (l <= 3.6e-72) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 1.44e+47) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.6e-72:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 1.44e+47:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.6e-72)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 1.44e+47)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * -2.0) * Float64(2.0 + Float64(n / Float64(Om / Float64(U - U_42_))))) * Float64(l * Float64(n * l))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.6e-72)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 1.44e+47)
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.6e-72], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.44e+47], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * -2.0), $MachinePrecision] * N[(2.0 + N[(n / N[(Om / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.6e-72

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.1%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.3%

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

   if 3.6e-72 < l < 1.4399999999999999e47

   1. Initial program 57.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 55.4%

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

   if 1.4399999999999999e47 < l

   1. Initial program 39.8%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   7. Simplified 38.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right)\right), Om\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(2 - n \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right), Om\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U \cdot -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{U* - U}{Om}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{1}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(\frac{n}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U* - U}\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U* - U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\left(n \cdot \ell\right) \cdot \ell\right)\right), Om\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\ell \cdot \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right)\right), Om\right)\right) \]

      18. *-lowering-*.f64 53.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right)\right), Om\right)\right) \]

   9. Applied egg-rr 53.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.44 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \ell \cdot \left(U* \cdot \frac{n}{Om}\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.4e-72)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= l 6.2e+46)
     (sqrt (* (* 2.0 U) (* n (- t (* 2.0 (/ (* l l) Om))))))
     (sqrt
      (* n (* (* 2.0 (* U l)) (/ (+ (* l -2.0) (* l (* U* (/ n Om)))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.4e-72) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 6.2e+46) {
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * l)) * (((l * -2.0) + (l * (U_42_ * (n / Om)))) / 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 <= 3.4d-72) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 6.2d+46) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * l)) * (((l * (-2.0d0)) + (l * (u_42 * (n / om)))) / 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 <= 3.4e-72) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 6.2e+46) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * l)) * (((l * -2.0) + (l * (U_42_ * (n / Om)))) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.4e-72:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 6.2e+46:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * l)) * (((l * -2.0) + (l * (U_42_ * (n / Om)))) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.4e-72)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 6.2e+46)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * l)) * Float64(Float64(Float64(l * -2.0) + Float64(l * Float64(U_42_ * Float64(n / Om)))) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.4e-72)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 6.2e+46)
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((n * ((2.0 * (U * l)) * (((l * -2.0) + (l * (U_42_ * (n / Om)))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.4e-72], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 6.2e+46], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * -2.0), $MachinePrecision] + N[(l * N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.3999999999999998e-72

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.1%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.3%

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

   if 3.3999999999999998e-72 < l < 6.1999999999999995e46

   1. Initial program 57.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 55.4%

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

   if 6.1999999999999995e46 < l

   1. Initial program 39.8%

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

   3. Simplified 56.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 60.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 60.9%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(U \cdot \ell\right) \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(U \cdot \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), Om\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       9. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right), Om\right), n\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right), Om\right), n\right)\right) \]

       12. *-lowering-*.f64 46.0%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \ell\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 46.0%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \left(-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}\right)}{Om}\right), n\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(U \cdot \ell\right)\right), \left(\frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right)\right), n\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \ell\right)\right), \left(\frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right)\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \left(\frac{-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}}{Om}\right)\right), n\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\left(-2 \cdot \ell + U* \cdot \frac{\ell \cdot n}{Om}\right), Om\right)\right), n\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \ell\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right), Om\right)\right), n\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\ell \cdot -2\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right), Om\right)\right), n\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right), Om\right)\right), n\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(\frac{\ell \cdot n}{Om} \cdot U*\right)\right), Om\right)\right), n\right)\right) \]

       11. associate-/l* N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot U*\right)\right), Om\right)\right), n\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(\ell \cdot \left(\frac{n}{Om} \cdot U*\right)\right)\right), Om\right)\right), n\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om} \cdot U*\right)\right)\right), Om\right)\right), n\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{n}{Om}\right), U*\right)\right)\right), Om\right)\right), n\right)\right) \]

       15. /-lowering-/.f64 54.8%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, Om\right), U*\right)\right)\right), Om\right)\right), n\right)\right) \]

   14. Applied egg-rr 54.8%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot \left(U \cdot \ell\right)\right) \cdot \frac{\ell \cdot -2 + \ell \cdot \left(U* \cdot \frac{n}{Om}\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.05e+153)
   (sqrt
    (*
     n
     (*
      U
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (sqrt
    (/ (* (* (* U -2.0) (+ 2.0 (/ n (/ Om (- U U*))))) (* l (* n l))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.05e+153) {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * 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 (l <= 1.05d+153) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt(((((u * (-2.0d0)) * (2.0d0 + (n / (om / (u - u_42))))) * (l * (n * 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 (l <= 1.05e+153) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = Math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.05e+153:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.05e+153)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l)))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * -2.0) * Float64(2.0 + Float64(n / Float64(Om / Float64(U - U_42_))))) * Float64(l * Float64(n * l))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.05e+153)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.05e+153], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * -2.0), $MachinePrecision] * N[(2.0 + N[(n / N[(Om / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05000000000000008e153

   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 58.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 60.9%

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

   if 1.05000000000000008e153 < l

   1. Initial program 20.5%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   7. Simplified 27.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right)\right), Om\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(2 - n \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right), Om\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U \cdot -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{U* - U}{Om}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{1}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(\frac{n}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U* - U}\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U* - U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\left(n \cdot \ell\right) \cdot \ell\right)\right), Om\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\ell \cdot \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right)\right), Om\right)\right) \]

      18. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right)\right), Om\right)\right) \]

   9. Applied egg-rr 54.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -48000000000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -48000000000000.0)
   (sqrt
    (*
     2.0
     (*
      (+ t (* (/ l Om) (+ (* l -2.0) (* (/ (* n l) Om) (- U* U)))))
      (* n U))))
   (sqrt
    (*
     n
     (* U (* 2.0 (+ t (/ (+ (* l -2.0) (* U* (* l (/ n Om)))) (/ Om l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -48000000000000.0) {
		tmp = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (((n * l) / Om) * (U_42_ - U))))) * (n * U))));
	} else {
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / 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 (u <= (-48000000000000.0d0)) then
        tmp = sqrt((2.0d0 * ((t + ((l / om) * ((l * (-2.0d0)) + (((n * l) / om) * (u_42 - u))))) * (n * u))))
    else
        tmp = sqrt((n * (u * (2.0d0 * (t + (((l * (-2.0d0)) + (u_42 * (l * (n / om)))) / (om / 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 (U <= -48000000000000.0) {
		tmp = Math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (((n * l) / Om) * (U_42_ - U))))) * (n * U))));
	} else {
		tmp = Math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -48000000000000.0:
		tmp = math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (((n * l) / Om) * (U_42_ - U))))) * (n * U))))
	else:
		tmp = math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -48000000000000.0)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(Float64(Float64(n * l) / Om) * Float64(U_42_ - U))))) * Float64(n * U))));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(l * Float64(n / Om)))) / Float64(Om / l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -48000000000000.0)
		tmp = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (((n * l) / Om) * (U_42_ - U))))) * (n * U))));
	else
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -48000000000000.0], N[Sqrt[N[(2.0 * N[(N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -4.8e13

   1. Initial program 54.1%

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

   3. Simplified 62.9%

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

   if -4.8e13 < U

   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 56.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 62.0%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 61.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 61.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -48000000000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.15 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.15e-72)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= l 3.35e+47)
     (sqrt (* (* 2.0 U) (* n (- t (* 2.0 (/ (* l l) Om))))))
     (sqrt (* n (/ (* -2.0 (* (* U (* l l)) (- 2.0 (* U* (/ n Om))))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.15e-72) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 3.35e+47) {
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((n * ((-2.0 * ((U * (l * l)) * (2.0 - (U_42_ * (n / Om))))) / 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 <= 2.15d-72) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 3.35d+47) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((n * (((-2.0d0) * ((u * (l * l)) * (2.0d0 - (u_42 * (n / om))))) / 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 <= 2.15e-72) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 3.35e+47) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((n * ((-2.0 * ((U * (l * l)) * (2.0 - (U_42_ * (n / Om))))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.15e-72:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 3.35e+47:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((n * ((-2.0 * ((U * (l * l)) * (2.0 - (U_42_ * (n / Om))))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.15e-72)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 3.35e+47)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l * l)) * Float64(2.0 - Float64(U_42_ * Float64(n / Om))))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.15e-72)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 3.35e+47)
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((n * ((-2.0 * ((U * (l * l)) * (2.0 - (U_42_ * (n / Om))))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.15e-72], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 3.35e+47], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(-2.0 * N[(N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.1499999999999999e-72

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.1%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.3%

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

   if 2.1499999999999999e-72 < l < 3.34999999999999986e47

   1. Initial program 57.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 55.4%

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

   if 3.34999999999999986e47 < l

   1. Initial program 39.8%

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

   3. Simplified 56.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 60.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 60.9%

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

   10. Taylor expanded in l around -inf

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)}{Om}\right)}, n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(2 + \left(\mathsf{neg}\left(\frac{U* \cdot n}{Om}\right)\right)\right)\right)\right)}{Om}\right), n\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right)}{Om}\right), n\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right), n\right)\right) \]

   12. Simplified 43.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.15 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e-72)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= l 1.6e+47)
     (sqrt (* (* 2.0 U) (* n (- t (* 2.0 (/ (* l l) Om))))))
     (sqrt (* n (* U (/ (* -2.0 (* (* l l) (- 2.0 (* U* (/ n Om))))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e-72) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 1.6e+47) {
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((n * (U * ((-2.0 * ((l * l) * (2.0 - (U_42_ * (n / Om))))) / 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 <= 2d-72) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (l <= 1.6d+47) then
        tmp = sqrt(((2.0d0 * u) * (n * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((n * (u * (((-2.0d0) * ((l * l) * (2.0d0 - (u_42 * (n / om))))) / 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 <= 2e-72) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (l <= 1.6e+47) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((n * (U * ((-2.0 * ((l * l) * (2.0 - (U_42_ * (n / Om))))) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2e-72:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif l <= 1.6e+47:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((n * (U * ((-2.0 * ((l * l) * (2.0 - (U_42_ * (n / Om))))) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2e-72)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (l <= 1.6e+47)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(-2.0 * Float64(Float64(l * l) * Float64(2.0 - Float64(U_42_ * Float64(n / Om))))) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2e-72)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (l <= 1.6e+47)
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((n * (U * ((-2.0 * ((l * l) * (2.0 - (U_42_ * (n / Om))))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2e-72], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.6e+47], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(N[(-2.0 * N[(N[(l * l), $MachinePrecision] * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.9999999999999999e-72

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.1%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.3%

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

   if 1.9999999999999999e-72 < l < 1.6e47

   1. Initial program 57.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 55.4%

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

   if 1.6e47 < l

   1. Initial program 39.8%

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

   3. Simplified 56.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 60.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 60.9%

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

   10. Taylor expanded in l around -inf

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)}, U\right), n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)\right)}{Om}\right), U\right), n\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left({\ell}^{2} \cdot \left(2 + \left(\mathsf{neg}\left(\frac{U* \cdot n}{Om}\right)\right)\right)\right)}{Om}\right), U\right), n\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left({\ell}^{2} \cdot \left(2 - \frac{U* \cdot n}{Om}\right)\right)}{Om}\right), U\right), n\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left({\ell}^{2} \cdot \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2} \cdot \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(2 - \frac{U* \cdot n}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \left(\frac{U* \cdot n}{Om}\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

       10. associate-/l* N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \left(U* \cdot \frac{n}{Om}\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(U*, \left(\frac{n}{Om}\right)\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

       12. /-lowering-/.f64 41.0%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(n, Om\right)\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

   12. Simplified 41.0%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{-2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.7 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -1.7e+52)
   (sqrt
    (*
     (+ t (/ (+ (* l -2.0) (* U* (/ (* n l) Om))) (/ Om l)))
     (* 2.0 (* n U))))
   (sqrt
    (*
     n
     (* U (* 2.0 (+ t (/ (+ (* l -2.0) (* U* (* l (/ n Om)))) (/ Om l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1.7e+52) {
		tmp = sqrt(((t + (((l * -2.0) + (U_42_ * ((n * l) / Om))) / (Om / l))) * (2.0 * (n * U))));
	} else {
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / 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 (u <= (-1.7d+52)) then
        tmp = sqrt(((t + (((l * (-2.0d0)) + (u_42 * ((n * l) / om))) / (om / l))) * (2.0d0 * (n * u))))
    else
        tmp = sqrt((n * (u * (2.0d0 * (t + (((l * (-2.0d0)) + (u_42 * (l * (n / om)))) / (om / 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 (U <= -1.7e+52) {
		tmp = Math.sqrt(((t + (((l * -2.0) + (U_42_ * ((n * l) / Om))) / (Om / l))) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -1.7e+52:
		tmp = math.sqrt(((t + (((l * -2.0) + (U_42_ * ((n * l) / Om))) / (Om / l))) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -1.7e+52)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om))) / Float64(Om / l))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(l * Float64(n / Om)))) / Float64(Om / l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -1.7e+52)
		tmp = sqrt(((t + (((l * -2.0) + (U_42_ * ((n * l) / Om))) / (Om / l))) * (2.0 * (n * U))));
	else
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -1.7e+52], N[Sqrt[N[(N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1.7e52

   1. Initial program 51.9%

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

   3. Simplified 59.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 46.7%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 46.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 46.7%

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

       1. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot 2\right) \cdot \left(U \cdot n\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{U* \cdot \left(\ell \cdot \frac{n}{Om}\right) + \ell \cdot -2}{\frac{Om}{\ell}}\right), \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

   11. Applied egg-rr 59.7%

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

   if -1.7e52 < U

   1. Initial program 50.3%

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

   3. Simplified 57.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 62.6%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 62.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 62.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.7 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.6e+154)
   (sqrt
    (*
     n
     (* U (* 2.0 (+ t (/ (+ (* l -2.0) (* U* (* l (/ n Om)))) (/ Om l)))))))
   (sqrt
    (/ (* (* (* U -2.0) (+ 2.0 (/ n (/ Om (- U U*))))) (* l (* n l))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.6e+154) {
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else {
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * 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 (l <= 6.6d+154) then
        tmp = sqrt((n * (u * (2.0d0 * (t + (((l * (-2.0d0)) + (u_42 * (l * (n / om)))) / (om / l)))))))
    else
        tmp = sqrt(((((u * (-2.0d0)) * (2.0d0 + (n / (om / (u - u_42))))) * (l * (n * 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 (l <= 6.6e+154) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else {
		tmp = Math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.6e+154:
		tmp = math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))))
	else:
		tmp = math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.6e+154)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(l * Float64(n / Om)))) / Float64(Om / l)))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * -2.0) * Float64(2.0 + Float64(n / Float64(Om / Float64(U - U_42_))))) * Float64(l * Float64(n * l))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.6e+154)
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	else
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.6e+154], N[Sqrt[N[(n * N[(U * N[(2.0 * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * -2.0), $MachinePrecision] * N[(2.0 + N[(n / N[(Om / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.6e154

   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 58.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 60.9%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot \frac{n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \left(\frac{n}{Om}\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

      5. /-lowering-/.f64 60.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(n, Om\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), U\right), n\right)\right) \]

   9. Simplified 60.5%

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

   if 6.6e154 < l

   1. Initial program 20.5%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   7. Simplified 27.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right)\right), Om\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(2 - n \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right), Om\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U \cdot -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{U* - U}{Om}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{1}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(\frac{n}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U* - U}\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U* - U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\left(n \cdot \ell\right) \cdot \ell\right)\right), Om\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\ell \cdot \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right)\right), Om\right)\right) \]

      18. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right)\right), Om\right)\right) \]

   9. Applied egg-rr 54.7%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.4e+80)
   (sqrt
    (*
     2.0
     (* (* n U) (+ t (* (/ l Om) (+ (* l -2.0) (* U* (/ (* n l) Om))))))))
   (sqrt
    (/ (* (* (* U -2.0) (+ 2.0 (/ n (/ Om (- U U*))))) (* l (* n l))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.4e+80) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))));
	} else {
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * 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 (l <= 8.4d+80) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l / om) * ((l * (-2.0d0)) + (u_42 * ((n * l) / om))))))))
    else
        tmp = sqrt(((((u * (-2.0d0)) * (2.0d0 + (n / (om / (u - u_42))))) * (l * (n * 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 (l <= 8.4e+80) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))));
	} else {
		tmp = Math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.4e+80:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))))
	else:
		tmp = math.sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.4e+80)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om))))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * -2.0) * Float64(2.0 + Float64(n / Float64(Om / Float64(U - U_42_))))) * Float64(l * Float64(n * l))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8.4e+80)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))))));
	else
		tmp = sqrt(((((U * -2.0) * (2.0 + (n / (Om / (U - U_42_))))) * (l * (n * l))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.4e+80], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * -2.0), $MachinePrecision] * N[(2.0 + N[(n / N[(Om / N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.40000000000000005e80

   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 57.5%

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

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\left(U* \cdot \frac{\ell \cdot n}{Om}\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

      4. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right)\right)\right), \mathsf{*.f64}\left(n, U\right)\right)\right)\right) \]

   7. Simplified 57.5%

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

   if 8.40000000000000005e80 < l

   1. Initial program 34.6%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   7. Simplified 36.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right)\right), Om\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(2 - n \cdot \frac{U* - U}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right), Om\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U \cdot -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \left(2 - n \cdot \frac{U* - U}{Om}\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{U* - U}{Om}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(n \cdot \frac{1}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \left(\frac{n}{\frac{Om}{U* - U}}\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \left(\frac{Om}{U* - U}\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \left(U* - U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), Om\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\left(n \cdot \ell\right) \cdot \ell\right)\right), Om\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \left(\ell \cdot \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(n \cdot \ell\right)\right)\right), Om\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \left(\ell \cdot n\right)\right)\right), Om\right)\right) \]

      18. *-lowering-*.f64 53.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, -2\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \mathsf{\_.f64}\left(U*, U\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, n\right)\right)\right), Om\right)\right) \]

   9. Applied egg-rr 53.7%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U \cdot -2\right) \cdot \left(2 + \frac{n}{\frac{Om}{U - U*}}\right)\right) \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.2e-72)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (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_) {
	double tmp;
	if (l <= 4.2e-72) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt(((2.0 * U) * (n * (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 (l <= 4.2d-72) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * u) * (n * (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 (l <= 4.2e-72) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.2e-72:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.2e-72)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.2e-72)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * U) * (n * (t - (2.0 * ((l * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.2e-72], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2e-72

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.1%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.3%

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

   if 4.2e-72 < l

   1. Initial program 47.6%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 41.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 41.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+51}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{\left(\ell \cdot \ell\right) \cdot -4}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3e+51)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* n (* U (/ (* (* l l) -4.0) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3e+51) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((n * (U * (((l * l) * -4.0) / 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+51) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((n * (u * (((l * l) * (-4.0d0)) / 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+51) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((n * (U * (((l * l) * -4.0) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3e+51:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((n * (U * (((l * l) * -4.0) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3e+51)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(l * l) * -4.0) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3e+51)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((n * (U * (((l * l) * -4.0) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3e+51], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(N[(N[(l * l), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3e51

   1. Initial program 52.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.3%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.2%

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

   if 3e51 < l

   1. Initial program 38.2%

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

   3. Simplified 55.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in l around -inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}\right)}, U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2 \cdot \left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}{Om}\right), U\right), n\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left({\ell}^{2} \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(2 + \left(\mathsf{neg}\left(\frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(2 - \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right), Om\right), U\right), n\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \left(\frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(n \cdot \left(U* - U\right)\right), Om\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U* - U\right)\right), Om\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

      12. --lowering--.f64 39.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U*, U\right)\right), Om\right)\right)\right)\right), Om\right), U\right), n\right)\right) \]

   9. Simplified 39.4%

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

   10. Taylor expanded in n around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right)}, U\right), n\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), U\right), n\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), U\right), n\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), U\right), n\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), U\right), n\right)\right) \]

       5. *-lowering-*.f64 25.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), U\right), n\right)\right) \]

   12. Simplified 25.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+51}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \frac{\left(\ell \cdot \ell\right) \cdot -4}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.5e-81)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (pow (* (* 2.0 U) (* n t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8.5e-81) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = pow(((2.0 * U) * (n * t)), 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 <= 8.5d-81) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = ((2.0d0 * u) * (n * t)) ** 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 <= 8.5e-81) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.5e-81:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.5e-81)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8.5e-81)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.5e-81], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.5000000000000001e-81

   1. Initial program 51.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.3%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.5%

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

   if 8.5000000000000001e-81 < l

   1. Initial program 47.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 27.9%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto {t}^{\frac{1}{2}} \cdot {\left(\left(n \cdot 2\right) \cdot U\right)}^{\frac{1}{2}} \]

      5. associate-*l* N/A

         \[\leadsto {t}^{\frac{1}{2}} \cdot {\left(n \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}} \]

      6. unpow-prod-down N/A

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

      7. associate-*l* N/A

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

      8. unpow-prod-down N/A

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

      9. pow-prod-down N/A

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

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\left(t \cdot n\right) \cdot \left(2 \cdot U\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(t \cdot n\right), \left(2 \cdot U\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, n\right), \left(2 \cdot U\right)\right), \frac{1}{2}\right) \]

      13. *-lowering-*.f64 30.6%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, n\right), \mathsf{*.f64}\left(2, U\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 30.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.6e-72)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.6e-72) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	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 <= 5.6d-72) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    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 <= 5.6e-72) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.6e-72:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.6e-72)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.6e-72)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.6e-72], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5999999999999996e-72

   1. Initial program 51.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.1%

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

      1. pow1/2 N/A

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

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(U \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \left(t \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. *-lowering-*.f64 47.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 47.3%

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

   if 5.5999999999999996e-72 < l

   1. Initial program 47.6%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 30.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 30.8%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-72}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.4e-81) (sqrt (* n (* U (* 2.0 t)))) (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.4e-81) {
		tmp = sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	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 <= 3.4d-81) then
        tmp = sqrt((n * (u * (2.0d0 * t))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    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 <= 3.4e-81) {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.4e-81:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.4e-81)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.4e-81)
		tmp = sqrt((n * (U * (2.0 * t))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.4e-81], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3999999999999999e-81

   1. Initial program 51.9%

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

   3. Simplified 57.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(n \cdot U\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot \left(U \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)\right) \cdot U\right), n\right)\right) \]

   6. Applied egg-rr 60.3%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot t\right)}, U\right), n\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 46.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right)\right) \]

   9. Simplified 46.4%

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

   if 3.3999999999999999e-81 < l

   1. Initial program 47.0%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 30.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 30.5%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.2e-71) (sqrt (* (* (* 2.0 n) U) t)) (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.2e-71) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	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 <= 3.2d-71) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    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 <= 3.2e-71) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.2e-71:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.2e-71)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.2e-71)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.2e-71], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.1999999999999999e-71

   1. Initial program 51.4%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 44.9%

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

   if 3.1999999999999999e-71 < l

   1. Initial program 48.2%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 31.2%

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

Alternative 19: 35.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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))))
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))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

   2. *-lowering-*.f64 40.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

7. Simplified 40.6%

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

Reproduce

herbie shell --seed 2024141 
(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*))))))