Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 65.0%

Time: 21.7s

Alternatives: 24

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: 49.8% 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: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\\ \mathbf{if}\;n \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(n \cdot 2\right)}^{0.5} \cdot \sqrt{U \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))
   (if (<= n 3.3e-307)
     (sqrt (* U (* n (* 2.0 t_1))))
     (* (pow (* n 2.0) 0.5) (sqrt (* U t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l));
	double tmp;
	if (n <= 3.3e-307) {
		tmp = sqrt((U * (n * (2.0 * t_1))));
	} else {
		tmp = pow((n * 2.0), 0.5) * sqrt((U * t_1));
	}
	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 = t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l))
    if (n <= 3.3d-307) then
        tmp = sqrt((u * (n * (2.0d0 * t_1))))
    else
        tmp = ((n * 2.0d0) ** 0.5d0) * sqrt((u * t_1))
    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 = t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l));
	double tmp;
	if (n <= 3.3e-307) {
		tmp = Math.sqrt((U * (n * (2.0 * t_1))));
	} else {
		tmp = Math.pow((n * 2.0), 0.5) * Math.sqrt((U * t_1));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l))
	tmp = 0
	if n <= 3.3e-307:
		tmp = math.sqrt((U * (n * (2.0 * t_1))))
	else:
		tmp = math.pow((n * 2.0), 0.5) * math.sqrt((U * t_1))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t + Float64(Float64(Float64(Float64(U_42_ - U) / Float64(Float64(Om / l) / n)) + Float64(l * -2.0)) / Float64(Om / l)))
	tmp = 0.0
	if (n <= 3.3e-307)
		tmp = sqrt(Float64(U * Float64(n * Float64(2.0 * t_1))));
	else
		tmp = Float64((Float64(n * 2.0) ^ 0.5) * sqrt(Float64(U * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l));
	tmp = 0.0;
	if (n <= 3.3e-307)
		tmp = sqrt((U * (n * (2.0 * t_1))));
	else
		tmp = ((n * 2.0) ^ 0.5) * sqrt((U * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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]}, If[LessEqual[n, 3.3e-307], N[Sqrt[N[(U * N[(n * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(n * 2.0), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(U * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 3.3e-307

   1. Initial program 48.3%

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

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. 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 59.8%

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

   if 3.3e-307 < n

   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 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. 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 60.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

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

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

      11. *-commutative N/A

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

   8. Applied egg-rr 70.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(n \cdot 2\right)}^{0.5} \cdot \sqrt{U \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\\ \mathbf{if}\;n \leq 3.5 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t\_1} \cdot \sqrt{n}\\ \end{array} \end{array} \]

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))))))
   (if (<= n 3.5e-307) (sqrt (* U (* n t_1))) (* (sqrt (* U t_1)) (sqrt n)))))

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 tmp;
	if (n <= 3.5e-307) {
		tmp = sqrt((U * (n * t_1)));
	} else {
		tmp = sqrt((U * t_1)) * sqrt(n);
	}
	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 = 2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))
    if (n <= 3.5d-307) then
        tmp = sqrt((u * (n * t_1)))
    else
        tmp = sqrt((u * t_1)) * sqrt(n)
    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 = 2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)));
	double tmp;
	if (n <= 3.5e-307) {
		tmp = Math.sqrt((U * (n * t_1)));
	} else {
		tmp = Math.sqrt((U * t_1)) * Math.sqrt(n);
	}
	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)))
	tmp = 0
	if n <= 3.5e-307:
		tmp = math.sqrt((U * (n * t_1)))
	else:
		tmp = math.sqrt((U * t_1)) * math.sqrt(n)
	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))))
	tmp = 0.0
	if (n <= 3.5e-307)
		tmp = sqrt(Float64(U * Float64(n * t_1)));
	else
		tmp = Float64(sqrt(Float64(U * t_1)) * sqrt(n));
	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)));
	tmp = 0.0;
	if (n <= 3.5e-307)
		tmp = sqrt((U * (n * t_1)));
	else
		tmp = sqrt((U * t_1)) * sqrt(n);
	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]}, If[LessEqual[n, 3.5e-307], N[Sqrt[N[(U * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 3.5000000000000002e-307

   1. Initial program 48.3%

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

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. 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 59.8%

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

   if 3.5000000000000002e-307 < n

   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 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. 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 70.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.5 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 3.3e-307)
   (sqrt
    (*
     U
     (*
      n
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (*
    (sqrt n)
    (sqrt
     (* U (* 2.0 (+ t (/ (+ (* l -2.0) (/ (* U* (* n l)) Om)) (/ Om l)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 3.3e-307) {
		tmp = sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = sqrt(n) * sqrt((U * (2.0 * (t + (((l * -2.0) + ((U_42_ * (n * l)) / 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 (n <= 3.3d-307) then
        tmp = sqrt((u * (n * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt(n) * sqrt((u * (2.0d0 * (t + (((l * (-2.0d0)) + ((u_42 * (n * l)) / 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 (n <= 3.3e-307) {
		tmp = Math.sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt((U * (2.0 * (t + (((l * -2.0) + ((U_42_ * (n * l)) / Om)) / (Om / l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 3.3e-307:
		tmp = math.sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt(n) * math.sqrt((U * (2.0 * (t + (((l * -2.0) + ((U_42_ * (n * l)) / Om)) / (Om / l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 3.3e-307)
		tmp = sqrt(Float64(U * Float64(n * 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 = Float64(sqrt(n) * sqrt(Float64(U * Float64(2.0 * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(U_42_ * Float64(n * l)) / Om)) / Float64(Om / l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 3.3e-307)
		tmp = sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt(n) * sqrt((U * (2.0 * (t + (((l * -2.0) + ((U_42_ * (n * l)) / Om)) / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 3.3e-307], N[Sqrt[N[(U * N[(n * 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[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(U * N[(2.0 * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.3e-307

   1. Initial program 48.3%

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

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. 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 59.8%

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

   if 3.3e-307 < n

   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 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. 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 70.5%

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

   7. Taylor expanded in U* around inf

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

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

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

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

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

      3. *-lowering-*.f64 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell \cdot -2 + \frac{\ell \cdot \left(n \cdot U*\right)}{Om}}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 5.2e-305)
   (sqrt
    (*
     U
     (*
      n
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (*
    (sqrt (* n 2.0))
    (sqrt (* U (+ t (/ (+ (* l -2.0) (/ (* l (* n U*)) Om)) (/ Om l))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 5.2e-305) {
		tmp = sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((U * (t + (((l * -2.0) + ((l * (n * U_42_)) / 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 (n <= 5.2d-305) then
        tmp = sqrt((u * (n * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt((n * 2.0d0)) * sqrt((u * (t + (((l * (-2.0d0)) + ((l * (n * u_42)) / 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 (n <= 5.2e-305) {
		tmp = Math.sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((U * (t + (((l * -2.0) + ((l * (n * U_42_)) / Om)) / (Om / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 5.2e-305:
		tmp = math.sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((U * (t + (((l * -2.0) + ((l * (n * U_42_)) / Om)) / (Om / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 5.2e-305)
		tmp = sqrt(Float64(U * Float64(n * 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 = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(l * Float64(n * U_42_)) / Om)) / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 5.2e-305)
		tmp = sqrt((U * (n * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt((n * 2.0)) * sqrt((U * (t + (((l * -2.0) + ((l * (n * U_42_)) / Om)) / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 5.2e-305], N[Sqrt[N[(U * N[(n * 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[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(l * N[(n * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.2000000000000003e-305

   1. Initial program 48.3%

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

   3. Simplified 54.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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. 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 59.8%

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

   if 5.2000000000000003e-305 < n

   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 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. 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 60.4%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. *-lowering-*.f64 58.2%

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

   9. Simplified 58.2%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. pow1/2 N/A

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

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

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

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

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

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

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

       9. pow1/2 N/A

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

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

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

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

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

   11. Applied egg-rr 65.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t + \frac{\ell \cdot -2 + \frac{\ell \cdot \left(n \cdot U*\right)}{Om}}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 2e+56)
   (sqrt
    (*
     n
     (*
      U
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (* (sqrt (* t (* n 2.0))) (sqrt U))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 2e+56) {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = sqrt((t * (n * 2.0))) * sqrt(U);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 2d+56) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt((t * (n * 2.0d0))) * sqrt(u)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 2e+56) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = Math.sqrt((t * (n * 2.0))) * Math.sqrt(U);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 2e+56:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt((t * (n * 2.0))) * math.sqrt(U)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 2e+56)
		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 = Float64(sqrt(Float64(t * Float64(n * 2.0))) * sqrt(U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 2e+56)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt((t * (n * 2.0))) * sqrt(U);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 2e+56], 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[(N[Sqrt[N[(t * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 2.00000000000000018e56

   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 56.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 61.7%

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

   if 2.00000000000000018e56 < U

   1. Initial program 58.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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 48.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

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

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

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

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

      9. pow1/2 N/A

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

      10. sqrt-lowering-sqrt.f64 64.4%

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

   7. Applied egg-rr 64.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.6e+102)
   (sqrt
    (*
     n
     (*
      U
      (* 2.0 (+ t (/ (+ (/ (- U* U) (/ (/ Om l) n)) (* l -2.0)) (/ Om l)))))))
   (sqrt
    (* 2.0 (* (/ U (/ -1.0 (* l (- 2.0 (/ U* (/ Om n)))))) (/ n (/ Om l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.6e+102) {
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (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 (l <= 3.6d+102) then
        tmp = sqrt((n * (u * (2.0d0 * (t + ((((u_42 - u) / ((om / l) / n)) + (l * (-2.0d0))) / (om / l)))))))
    else
        tmp = sqrt((2.0d0 * ((u / ((-1.0d0) / (l * (2.0d0 - (u_42 / (om / n)))))) * (n / (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 (l <= 3.6e+102) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	} else {
		tmp = Math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.6e+102:
		tmp = math.sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))))
	else:
		tmp = math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.6e+102)
		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(2.0 * Float64(Float64(U / Float64(-1.0 / Float64(l * Float64(2.0 - Float64(U_42_ / Float64(Om / n)))))) * Float64(n / Float64(Om / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.6e+102)
		tmp = sqrt((n * (U * (2.0 * (t + ((((U_42_ - U) / ((Om / l) / n)) + (l * -2.0)) / (Om / l)))))));
	else
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.6e+102], 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[(2.0 * N[(N[(U / N[(-1.0 / N[(l * N[(2.0 - N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.6000000000000002e102

   1. Initial program 52.4%

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

   3. Simplified 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 60.7%

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

   if 3.6000000000000002e102 < l

   1. Initial program 28.7%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 36.4%

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

   10. Simplified 36.4%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 36.4%

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

   12. Applied egg-rr 36.4%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       15. /-lowering-/.f64 N/A

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

   14. Applied egg-rr 60.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.5e+103)
   (sqrt
    (*
     n
     (* U (* 2.0 (+ t (/ (+ (* l -2.0) (* U* (* l (/ n Om)))) (/ Om l)))))))
   (sqrt
    (* 2.0 (* (/ U (/ -1.0 (* l (- 2.0 (/ U* (/ Om n)))))) (/ n (/ Om l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.5e+103) {
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else {
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (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 (l <= 3.5d+103) then
        tmp = sqrt((n * (u * (2.0d0 * (t + (((l * (-2.0d0)) + (u_42 * (l * (n / om)))) / (om / l)))))))
    else
        tmp = sqrt((2.0d0 * ((u / ((-1.0d0) / (l * (2.0d0 - (u_42 / (om / n)))))) * (n / (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 (l <= 3.5e+103) {
		tmp = Math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	} else {
		tmp = Math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.5e+103:
		tmp = math.sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))))
	else:
		tmp = math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.5e+103)
		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(2.0 * Float64(Float64(U / Float64(-1.0 / Float64(l * Float64(2.0 - Float64(U_42_ / Float64(Om / n)))))) * Float64(n / Float64(Om / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.5e+103)
		tmp = sqrt((n * (U * (2.0 * (t + (((l * -2.0) + (U_42_ * (l * (n / Om)))) / (Om / l)))))));
	else
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.5e+103], 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[(2.0 * N[(N[(U / N[(-1.0 / N[(l * N[(2.0 - N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e103

   1. Initial program 52.4%

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

   3. Simplified 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 60.7%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. *-lowering-*.f64 58.9%

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

   9. Simplified 58.9%

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 59.3%

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

   11. Applied egg-rr 59.3%

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

   if 3.5e103 < l

   1. Initial program 28.7%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 36.4%

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

   10. Simplified 36.4%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 36.4%

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

   12. Applied egg-rr 36.4%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       15. /-lowering-/.f64 N/A

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

   14. Applied egg-rr 60.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(t + \frac{\ell \cdot -2 + U* \cdot \left(\ell \cdot \frac{n}{Om}\right)}{\frac{Om}{\ell}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.9e+103)
   (sqrt
    (*
     2.0
     (* (+ t (* (/ l Om) (+ (* l -2.0) (* U* (/ (* n l) Om))))) (* n U))))
   (sqrt
    (* 2.0 (* (/ U (/ -1.0 (* l (- 2.0 (/ U* (/ Om n)))))) (/ n (/ Om l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.9e+103) {
		tmp = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))) * (n * U))));
	} else {
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (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 (l <= 2.9d+103) then
        tmp = sqrt((2.0d0 * ((t + ((l / om) * ((l * (-2.0d0)) + (u_42 * ((n * l) / om))))) * (n * u))))
    else
        tmp = sqrt((2.0d0 * ((u / ((-1.0d0) / (l * (2.0d0 - (u_42 / (om / n)))))) * (n / (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 (l <= 2.9e+103) {
		tmp = Math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))) * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.9e+103:
		tmp = math.sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))) * (n * U))))
	else:
		tmp = math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.9e+103)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(U_42_ * Float64(Float64(n * l) / Om))))) * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U / Float64(-1.0 / Float64(l * Float64(2.0 - Float64(U_42_ / Float64(Om / n)))))) * Float64(n / Float64(Om / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.9e+103)
		tmp = sqrt((2.0 * ((t + ((l / Om) * ((l * -2.0) + (U_42_ * ((n * l) / Om))))) * (n * U))));
	else
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.9e+103], N[Sqrt[N[(2.0 * N[(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] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U / N[(-1.0 / N[(l * N[(2.0 - N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.8999999999999998e103

   1. Initial program 52.4%

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

   3. Simplified 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. 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 57.2%

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

   7. Simplified 57.2%

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

   if 2.8999999999999998e103 < l

   1. Initial program 28.7%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 36.4%

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

   10. Simplified 36.4%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 36.4%

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

   12. Applied egg-rr 36.4%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       15. /-lowering-/.f64 N/A

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

   14. Applied egg-rr 60.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.4e+38)
   (sqrt (* n (* U (+ (/ (* -4.0 (* l l)) Om) (* 2.0 t)))))
   (sqrt
    (* 2.0 (* (/ U (/ -1.0 (* l (- 2.0 (/ U* (/ Om n)))))) (/ n (/ Om l)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.4e+38) {
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (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 (l <= 2.4d+38) then
        tmp = sqrt((n * (u * ((((-4.0d0) * (l * l)) / om) + (2.0d0 * t)))))
    else
        tmp = sqrt((2.0d0 * ((u / ((-1.0d0) / (l * (2.0d0 - (u_42 / (om / n)))))) * (n / (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 (l <= 2.4e+38) {
		tmp = Math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = Math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.4e+38:
		tmp = math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))))
	else:
		tmp = math.sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.4e+38)
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(-4.0 * Float64(l * l)) / Om) + Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U / Float64(-1.0 / Float64(l * Float64(2.0 - Float64(U_42_ / Float64(Om / n)))))) * Float64(n / Float64(Om / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.4e+38)
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	else
		tmp = sqrt((2.0 * ((U / (-1.0 / (l * (2.0 - (U_42_ / (Om / n)))))) * (n / (Om / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.4e+38], N[Sqrt[N[(n * N[(U * N[(N[(N[(-4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U / N[(-1.0 / N[(l * N[(2.0 - N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.40000000000000017e38

   1. Initial program 50.6%

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

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 59.0%

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

   7. Taylor expanded in Om around inf

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 47.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \mathsf{*.f64}\left(2, t\right)\right), U\right), n\right)\right) \]

   9. Simplified 47.1%

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

   if 2.40000000000000017e38 < l

   1. Initial program 44.0%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 46.0%

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

   10. Simplified 46.0%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 46.1%

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

   12. Applied egg-rr 46.1%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

       5. neg-mul-1 N/A

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

       6. times-frac N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       15. /-lowering-/.f64 N/A

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

   14. Applied egg-rr 62.3%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\frac{U}{\frac{-1}{\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)}} \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(\left(\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.7e+39)
   (sqrt (* n (* U (+ (/ (* -4.0 (* l l)) Om) (* 2.0 t)))))
   (sqrt (* n (* U (* -2.0 (* (* l (- 2.0 (/ U* (/ Om n)))) (/ l Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.7e+39) {
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = sqrt((n * (U * (-2.0 * ((l * (2.0 - (U_42_ / (Om / 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 <= 1.7d+39) then
        tmp = sqrt((n * (u * ((((-4.0d0) * (l * l)) / om) + (2.0d0 * t)))))
    else
        tmp = sqrt((n * (u * ((-2.0d0) * ((l * (2.0d0 - (u_42 / (om / 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 <= 1.7e+39) {
		tmp = Math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = Math.sqrt((n * (U * (-2.0 * ((l * (2.0 - (U_42_ / (Om / n)))) * (l / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.7e+39:
		tmp = math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))))
	else:
		tmp = math.sqrt((n * (U * (-2.0 * ((l * (2.0 - (U_42_ / (Om / n)))) * (l / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.7e+39)
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(-4.0 * Float64(l * l)) / Om) + Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(-2.0 * Float64(Float64(l * Float64(2.0 - Float64(U_42_ / Float64(Om / n)))) * Float64(l / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.7e+39)
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	else
		tmp = sqrt((n * (U * (-2.0 * ((l * (2.0 - (U_42_ / (Om / n)))) * (l / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.7e+39], N[Sqrt[N[(n * N[(U * N[(N[(N[(-4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(-2.0 * N[(N[(l * N[(2.0 - N[(U$42$ / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6999999999999999e39

   1. Initial program 50.6%

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

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 59.0%

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

   7. Taylor expanded in Om around inf

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 47.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \mathsf{*.f64}\left(2, t\right)\right), U\right), n\right)\right) \]

   9. Simplified 47.1%

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

   if 1.6999999999999999e39 < l

   1. Initial program 44.0%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 46.0%

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

   10. Simplified 46.0%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 46.1%

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

   12. Applied egg-rr 46.1%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

   14. Applied egg-rr 52.6%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(-2 \cdot \left(\left(\ell \cdot \left(2 - \frac{U*}{\frac{Om}{n}}\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(n \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.5e+137)
   (sqrt (* n (* U (+ (/ (* -4.0 (* l l)) Om) (* 2.0 t)))))
   (sqrt (* n (/ (* 2.0 (* U (* n (* U* (* l l))))) (* Om Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.5e+137) {
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = sqrt((n * ((2.0 * (U * (n * (U_42_ * (l * l))))) / (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 <= 2.5d+137) then
        tmp = sqrt((n * (u * ((((-4.0d0) * (l * l)) / om) + (2.0d0 * t)))))
    else
        tmp = sqrt((n * ((2.0d0 * (u * (n * (u_42 * (l * l))))) / (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 <= 2.5e+137) {
		tmp = Math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = Math.sqrt((n * ((2.0 * (U * (n * (U_42_ * (l * l))))) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.5e+137:
		tmp = math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))))
	else:
		tmp = math.sqrt((n * ((2.0 * (U * (n * (U_42_ * (l * l))))) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.5e+137)
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(-4.0 * Float64(l * l)) / Om) + Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(n * Float64(Float64(2.0 * Float64(U * Float64(n * Float64(U_42_ * Float64(l * l))))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.5e+137)
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	else
		tmp = sqrt((n * ((2.0 * (U * (n * (U_42_ * (l * l))))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.5e+137], N[Sqrt[N[(n * N[(U * N[(N[(N[(-4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(2.0 * N[(U * N[(n * N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.5000000000000001e137

   1. Initial program 52.6%

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

   3. Simplified 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. 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 60.4%

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

   7. Taylor expanded in Om around inf

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 48.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \mathsf{*.f64}\left(2, t\right)\right), U\right), n\right)\right) \]

   9. Simplified 48.5%

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

   if 2.5000000000000001e137 < l

   1. Initial program 24.7%

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

   3. Simplified 49.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 52.8%

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

   7. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 39.0%

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

   9. Simplified 39.0%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{2 \cdot \left(U \cdot \left(n \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.5e+137)
   (sqrt (* n (* U (+ (/ (* -4.0 (* l l)) Om) (* 2.0 t)))))
   (sqrt (* (* U (* n 2.0)) (/ (* U* (* n (* l l))) (* Om Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.5e+137) {
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = sqrt(((U * (n * 2.0)) * ((U_42_ * (n * (l * l))) / (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 <= 2.5d+137) then
        tmp = sqrt((n * (u * ((((-4.0d0) * (l * l)) / om) + (2.0d0 * t)))))
    else
        tmp = sqrt(((u * (n * 2.0d0)) * ((u_42 * (n * (l * l))) / (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 <= 2.5e+137) {
		tmp = Math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = Math.sqrt(((U * (n * 2.0)) * ((U_42_ * (n * (l * l))) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.5e+137:
		tmp = math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))))
	else:
		tmp = math.sqrt(((U * (n * 2.0)) * ((U_42_ * (n * (l * l))) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.5e+137)
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(-4.0 * Float64(l * l)) / Om) + Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * Float64(Float64(U_42_ * Float64(n * Float64(l * l))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.5e+137)
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	else
		tmp = sqrt(((U * (n * 2.0)) * ((U_42_ * (n * (l * l))) / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.5e+137], N[Sqrt[N[(n * N[(U * N[(N[(N[(-4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(U$42$ * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.5000000000000001e137

   1. Initial program 52.6%

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

   3. Simplified 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. 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 60.4%

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

   7. Taylor expanded in Om around inf

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 48.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \mathsf{*.f64}\left(2, t\right)\right), U\right), n\right)\right) \]

   9. Simplified 48.5%

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

   if 2.5000000000000001e137 < l

   1. Initial program 24.7%

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

   3. Taylor expanded in U* around inf

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 38.8%

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

   5. Simplified 38.8%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n \cdot n}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.5e+137)
   (sqrt (* n (* U (+ (/ (* -4.0 (* l l)) Om) (* 2.0 t)))))
   (sqrt (* 2.0 (* U (* (/ (* U* (* l l)) Om) (/ (* n n) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.5e+137) {
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = sqrt((2.0 * (U * (((U_42_ * (l * l)) / Om) * ((n * n) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.5d+137) then
        tmp = sqrt((n * (u * ((((-4.0d0) * (l * l)) / om) + (2.0d0 * t)))))
    else
        tmp = sqrt((2.0d0 * (u * (((u_42 * (l * l)) / om) * ((n * n) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.5e+137) {
		tmp = Math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (((U_42_ * (l * l)) / Om) * ((n * n) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.5e+137:
		tmp = math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))))
	else:
		tmp = math.sqrt((2.0 * (U * (((U_42_ * (l * l)) / Om) * ((n * n) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.5e+137)
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(-4.0 * Float64(l * l)) / Om) + Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(Float64(Float64(U_42_ * Float64(l * l)) / Om) * Float64(Float64(n * n) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.5e+137)
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	else
		tmp = sqrt((2.0 * (U * (((U_42_ * (l * l)) / Om) * ((n * n) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.5e+137], N[Sqrt[N[(n * N[(U * N[(N[(N[(-4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(N[(U$42$ * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.5000000000000001e137

   1. Initial program 52.6%

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

   3. Simplified 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. 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 60.4%

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

   7. Taylor expanded in Om around inf

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 48.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \mathsf{*.f64}\left(2, t\right)\right), U\right), n\right)\right) \]

   9. Simplified 48.5%

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

   if 2.5000000000000001e137 < l

   1. Initial program 24.7%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 39.8%

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

   7. Simplified 39.8%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 39.8%

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

   10. Simplified 39.8%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

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

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 39.8%

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

   12. Applied egg-rr 39.8%

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

   13. Taylor expanded in U* around inf

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

       1. associate-/l* N/A

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

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

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. times-frac N/A

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

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

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

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

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

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

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

       9. unpow2 N/A

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

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

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 35.3%

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

   15. Simplified 35.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n \cdot n}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 5e-57)
   (sqrt (* n (* U (+ (/ (* -4.0 (* l l)) Om) (* 2.0 t)))))
   (sqrt (* (* U (* n 2.0)) (- t (* 2.0 (/ (* l l) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 5e-57) {
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = sqrt(((U * (n * 2.0)) * (t - (2.0 * ((l * l) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 5d-57) then
        tmp = sqrt((n * (u * ((((-4.0d0) * (l * l)) / om) + (2.0d0 * t)))))
    else
        tmp = sqrt(((u * (n * 2.0d0)) * (t - (2.0d0 * ((l * l) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 5e-57) {
		tmp = Math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	} else {
		tmp = Math.sqrt(((U * (n * 2.0)) * (t - (2.0 * ((l * l) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 5e-57:
		tmp = math.sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))))
	else:
		tmp = math.sqrt(((U * (n * 2.0)) * (t - (2.0 * ((l * l) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 5e-57)
		tmp = sqrt(Float64(n * Float64(U * Float64(Float64(Float64(-4.0 * Float64(l * l)) / Om) + Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * 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 (U <= 5e-57)
		tmp = sqrt((n * (U * (((-4.0 * (l * l)) / Om) + (2.0 * t)))));
	else
		tmp = sqrt(((U * (n * 2.0)) * (t - (2.0 * ((l * l) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 5e-57], N[Sqrt[N[(n * N[(U * N[(N[(N[(-4.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 5.0000000000000002e-57

   1. Initial program 44.5%

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

   3. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 58.4%

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

   7. Taylor expanded in Om around inf

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \frac{{\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-4 \cdot {\ell}^{2}}{Om}\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {\ell}^{2}\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left({\ell}^{2}\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(\ell \cdot \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \left(2 \cdot t\right)\right), U\right), n\right)\right) \]

      7. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(\ell, \ell\right)\right), Om\right), \mathsf{*.f64}\left(2, t\right)\right), U\right), n\right)\right) \]

   9. Simplified 42.8%

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

   if 5.0000000000000002e-57 < U

   1. Initial program 67.2%

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

   3. Taylor expanded in Om around inf

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

      1. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 65.4%

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

   5. Simplified 65.4%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{-4 \cdot \left(\ell \cdot \ell\right)}{Om} + 2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;U* \leq 9 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* 2.0 (/ (* l l) Om)))))
   (if (<= U* 9e-195)
     (sqrt (* (* U (* n 2.0)) t_1))
     (sqrt (* (* 2.0 U) (* n t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (2.0 * ((l * l) / Om));
	double tmp;
	if (U_42_ <= 9e-195) {
		tmp = sqrt(((U * (n * 2.0)) * t_1));
	} else {
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	}
	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 = t - (2.0d0 * ((l * l) / om))
    if (u_42 <= 9d-195) then
        tmp = sqrt(((u * (n * 2.0d0)) * t_1))
    else
        tmp = sqrt(((2.0d0 * u) * (n * t_1)))
    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 = t - (2.0 * ((l * l) / Om));
	double tmp;
	if (U_42_ <= 9e-195) {
		tmp = Math.sqrt(((U * (n * 2.0)) * t_1));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t_1)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (2.0 * ((l * l) / Om))
	tmp = 0
	if U_42_ <= 9e-195:
		tmp = math.sqrt(((U * (n * 2.0)) * t_1))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * t_1)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	tmp = 0.0
	if (U_42_ <= 9e-195)
		tmp = sqrt(Float64(Float64(U * Float64(n * 2.0)) * t_1));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - (2.0 * ((l * l) / Om));
	tmp = 0.0;
	if (U_42_ <= 9e-195)
		tmp = sqrt(((U * (n * 2.0)) * t_1));
	else
		tmp = sqrt(((2.0 * U) * (n * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$42$, 9e-195], N[Sqrt[N[(N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U* < 9e-195

   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 Om around inf

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

      1. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 46.2%

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

   5. Simplified 46.2%

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

   if 9e-195 < U*

   1. Initial program 45.6%

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

   3. Simplified 57.9%

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

   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 48.0%

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

   7. Simplified 48.0%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 9 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.5e-217)
   (sqrt (* n (* U (* 2.0 t))))
   (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 <= 8.5e-217) {
		tmp = sqrt((n * (U * (2.0 * t))));
	} 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 <= 8.5d-217) then
        tmp = sqrt((n * (u * (2.0d0 * t))))
    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 <= 8.5e-217) {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	} 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 <= 8.5e-217:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	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 <= 8.5e-217)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	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 <= 8.5e-217)
		tmp = sqrt((n * (U * (2.0 * t))));
	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, 8.5e-217], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 < 8.4999999999999994e-217

   1. Initial program 49.1%

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

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 57.8%

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

   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 36.2%

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

   9. Simplified 36.2%

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

   if 8.4999999999999994e-217 < l

   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 58.1%

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

   5. Taylor expanded in 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 48.0%

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

   7. Simplified 48.0%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8.8e+44)
   (sqrt (* n (* U (* 2.0 t))))
   (sqrt (* 2.0 (* (* n U) (/ (* -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 <= 8.8e+44) {
		tmp = sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = sqrt((2.0 * ((n * U) * ((-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 <= 8.8d+44) then
        tmp = sqrt((n * (u * (2.0d0 * t))))
    else
        tmp = sqrt((2.0d0 * ((n * u) * (((-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 <= 8.8e+44) {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = Math.sqrt((2.0 * ((n * U) * ((-2.0 * (l * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8.8e+44:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	else:
		tmp = math.sqrt((2.0 * ((n * U) * ((-2.0 * (l * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8.8e+44)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(Float64(-2.0 * 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 <= 8.8e+44)
		tmp = sqrt((n * (U * (2.0 * t))));
	else
		tmp = sqrt((2.0 * ((n * U) * ((-2.0 * (l * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8.8e+44], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.79999999999999983e44

   1. Initial program 50.4%

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

   3. Simplified 55.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 59.2%

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

   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 40.3%

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

   9. Simplified 40.3%

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

   if 8.79999999999999983e44 < l

   1. Initial program 44.8%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.6%

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

   10. Simplified 33.6%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \frac{-2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 7e+44)
   (sqrt (* n (* U (* 2.0 t))))
   (sqrt (* -4.0 (/ (* U (* n (* l l))) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 7e+44) {
		tmp = sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = sqrt((-4.0 * ((U * (n * (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 <= 7d+44) then
        tmp = sqrt((n * (u * (2.0d0 * t))))
    else
        tmp = sqrt(((-4.0d0) * ((u * (n * (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 <= 7e+44) {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = Math.sqrt((-4.0 * ((U * (n * (l * l))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 7e+44:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	else:
		tmp = math.sqrt((-4.0 * ((U * (n * (l * l))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 7e+44)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(n * 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 <= 7e+44)
		tmp = sqrt((n * (U * (2.0 * t))));
	else
		tmp = sqrt((-4.0 * ((U * (n * (l * l))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 7e+44], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.9999999999999998e44

   1. Initial program 50.4%

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

   3. Simplified 55.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 59.2%

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

   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 40.3%

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

   9. Simplified 40.3%

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

   if 6.9999999999999998e44 < l

   1. Initial program 44.8%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in U around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 44.9%

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

   10. Simplified 44.9%

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

   11. Taylor expanded in U* around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), Om\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot {\ell}^{2}\right)\right), Om\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left({\ell}^{2}\right)\right)\right), Om\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(\ell \cdot \ell\right)\right)\right), Om\right)\right)\right) \]

       7. *-lowering-*.f64 29.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), Om\right)\right)\right) \]

   13. Simplified 29.7%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.5e+44)
   (sqrt (* n (* U (* 2.0 t))))
   (sqrt (* -4.0 (* U (/ (* n (* l l)) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.5e+44) {
		tmp = sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = sqrt((-4.0 * (U * ((n * (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 <= 6.5d+44) then
        tmp = sqrt((n * (u * (2.0d0 * t))))
    else
        tmp = sqrt(((-4.0d0) * (u * ((n * (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 <= 6.5e+44) {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	} else {
		tmp = Math.sqrt((-4.0 * (U * ((n * (l * l)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.5e+44:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	else:
		tmp = math.sqrt((-4.0 * (U * ((n * (l * l)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.5e+44)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	else
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64(Float64(n * 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 <= 6.5e+44)
		tmp = sqrt((n * (U * (2.0 * t))));
	else
		tmp = sqrt((-4.0 * (U * ((n * (l * l)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.5e+44], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(U * N[(N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.50000000000000018e44

   1. Initial program 50.4%

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

   3. Simplified 55.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 59.2%

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

   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 40.3%

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

   9. Simplified 40.3%

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

   if 6.50000000000000018e44 < l

   1. Initial program 44.8%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in l around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left(\frac{{\ell}^{2} \cdot n}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot n\right), Om\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right), Om\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right), Om\right)\right)\right)\right) \]

      7. *-lowering-*.f64 29.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right), Om\right)\right)\right)\right) \]

   10. Simplified 29.9%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+117}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -3.4e+117)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (pow (* 2.0 (* n (* t U))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -3.4e+117) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = pow((2.0 * (n * (t * U))), 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 (t <= (-3.4d+117)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = (2.0d0 * (n * (t * u))) ** 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 (t <= -3.4e+117) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (n * (t * U))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -3.4e+117:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.pow((2.0 * (n * (t * U))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -3.4e+117)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(n * Float64(t * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -3.4e+117)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = (2.0 * (n * (t * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -3.4e+117], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.4000000000000001e117

   1. Initial program 48.8%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

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

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

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

      4. *-lowering-*.f64 39.7%

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

   7. Simplified 39.7%

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

      1. pow1/2 N/A

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 42.2%

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

   9. Applied egg-rr 42.2%

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

   if -3.4000000000000001e117 < t

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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 33.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
   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 36.9%

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

   7. Applied egg-rr 36.9%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+117}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.6e-181)
   (sqrt (* n (* U (* 2.0 t))))
   (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 <= 2.6e-181) {
		tmp = sqrt((n * (U * (2.0 * t))));
	} 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 <= 2.6d-181) then
        tmp = sqrt((n * (u * (2.0d0 * t))))
    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 <= 2.6e-181) {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	} 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 <= 2.6e-181:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	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 <= 2.6e-181)
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	else
		tmp = Float64(2.0 * Float64(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 <= 2.6e-181)
		tmp = sqrt((n * (U * (2.0 * t))));
	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, 2.6e-181], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.59999999999999999e-181

   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 54.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 57.5%

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

   7. Taylor expanded in 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 36.1%

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

   9. Simplified 36.1%

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

   if 2.59999999999999999e-181 < l

   1. Initial program 51.3%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

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

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

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

      4. *-lowering-*.f64 33.7%

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

   7. Simplified 33.7%

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

      1. pow1/2 N/A

         \[\leadsto {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(t \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, n\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 34.7%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -9e+117) (sqrt (* (* 2.0 U) (* n t))) (sqrt (* n (* U (* 2.0 t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -9e+117) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt((n * (U * (2.0 * 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 (t <= (-9d+117)) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt((n * (u * (2.0d0 * 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 (t <= -9e+117) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt((n * (U * (2.0 * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -9e+117:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt((n * (U * (2.0 * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -9e+117)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = sqrt(Float64(n * Float64(U * Float64(2.0 * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -9e+117)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt((n * (U * (2.0 * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -9e+117], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(U * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9e117

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 55.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 39.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 39.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if -9e117 < t

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   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 61.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(t + \frac{\frac{U* - U}{\frac{\frac{Om}{\ell}}{n}} + \ell \cdot -2}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot n}} \]

   7. Taylor expanded in 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 36.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, t\right), U\right), n\right)\right) \]

   9. Simplified 36.5%

      \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot t\right)} \cdot U\right) \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(2 \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 1.4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 1.4e-187)
   (sqrt (* t (* U (* n 2.0))))
   (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 (U_42_ <= 1.4e-187) {
		tmp = sqrt((t * (U * (n * 2.0))));
	} 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 (u_42 <= 1.4d-187) then
        tmp = sqrt((t * (u * (n * 2.0d0))))
    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 (U_42_ <= 1.4e-187) {
		tmp = Math.sqrt((t * (U * (n * 2.0))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 1.4e-187:
		tmp = math.sqrt((t * (U * (n * 2.0))))
	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 (U_42_ <= 1.4e-187)
		tmp = sqrt(Float64(t * Float64(U * Float64(n * 2.0))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 1.4e-187)
		tmp = sqrt((t * (U * (n * 2.0))));
	else
		tmp = sqrt(((2.0 * U) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 1.4e-187], N[Sqrt[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 1.4e-187

   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 38.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if 1.4e-187 < U*

   1. Initial program 45.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(n \cdot U\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 34.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   7. Simplified 34.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 1.4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 35.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

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(Float64(2.0 * 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[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.4%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 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. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 35.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

7. Simplified 35.1%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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*))))))