Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 69.1%

Time: 24.9s

Alternatives: 22

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 69.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot \ell\right) \cdot \left(\left(n \cdot \ell\right) \cdot -2\right)}{Om \cdot Om} + \frac{n \cdot \left(2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + U* \cdot t\_2\right)\right)\right)}{U}} \cdot \left|U\right|\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_2 \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot t\_2\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* U (* 2.0 n))
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_2 (/ (* n l) Om)))
   (if (<= t_1 0.0)
     (*
      (sqrt
       (+
        (/ (* (* n l) (* (* n l) -2.0)) (* Om Om))
        (/ (* n (* 2.0 (+ t (* (/ l Om) (+ (* l -2.0) (* U* t_2)))))) U)))
      (fabs U))
     (if (<= t_1 INFINITY)
       (sqrt
        (* (* 2.0 (* n U)) (+ t (* (/ l Om) (+ (* l -2.0) (* t_2 (- U* U)))))))
       (sqrt
        (-
         (* U (* t (* 2.0 n)))
         (* (* 2.0 (* U t_2)) (- (/ (- U U*) (/ Om (* n l))) (* l -2.0)))))))))

C

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

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_2 = (n * l) / Om;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((((n * l) * ((n * l) * -2.0)) / (Om * Om)) + ((n * (2.0 * (t + ((l / Om) * ((l * -2.0) + (U_42_ * t_2)))))) / U))) * Math.abs(U);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_2 * (U_42_ - U)))))));
	} else {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_2)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((U * (2.0 * n)) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	t_2 = (n * l) / Om
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((((n * l) * ((n * l) * -2.0)) / (Om * Om)) + ((n * (2.0 * (t + ((l / Om) * ((l * -2.0) + (U_42_ * t_2)))))) / U))) * math.fabs(U)
	elif t_1 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_2 * (U_42_ - U)))))))
	else:
		tmp = math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_2)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $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]}, Block[{t$95$2 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(N[(N[(N[(n * l), $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(2.0 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(U$42$ * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(U * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 8.6%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 8.6%

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

   5. Taylor expanded in U around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

   7. Simplified 11.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   9. Applied egg-rr 66.7%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 75.1%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 78.9%

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

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

   1. Initial program 0.0%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 27.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 46.8%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 65.3%

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

   9. Simplified 65.3%

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

4. Final simplification 75.0%

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

Alternative 2: 65.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t\_2 \cdot \left(U - U*\right)\right)\\ t_4 := \frac{Om}{n \cdot \ell}\\ \mathbf{if}\;t\_3 \leq 0.0001:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{t\_4}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t\_2 \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(\frac{U - U*}{t\_4} - \ell \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3 (* t_1 (- (- t (* 2.0 (/ (* l l) Om))) (* t_2 (- U U*)))))
        (t_4 (/ Om (* n l))))
   (if (<= t_3 0.0001)
     (sqrt
      (* (* 2.0 n) (* U (+ t (/ (+ (* l -2.0) (/ (- U* U) t_4)) (/ Om l))))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (+ (- t (* 2.0 (/ l (/ Om l)))) (* t_2 (- U* U)))))
       (sqrt
        (-
         (* U (* t (* 2.0 n)))
         (* (* 2.0 (* U (/ (* n l) Om))) (- (/ (- U U*) t_4) (* l -2.0)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = n * pow((l / Om), 2.0);
	double t_3 = t_1 * ((t - (2.0 * ((l * l) / Om))) - (t_2 * (U - U_42_)));
	double t_4 = Om / (n * l);
	double tmp;
	if (t_3 <= 0.0001) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_4)) / (Om / l))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (U_42_ - U)))));
	} else {
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * ((n * l) / Om))) * (((U - U_42_) / t_4) - (l * -2.0)))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = n * Math.pow((l / Om), 2.0);
	double t_3 = t_1 * ((t - (2.0 * ((l * l) / Om))) - (t_2 * (U - U_42_)));
	double t_4 = Om / (n * l);
	double tmp;
	if (t_3 <= 0.0001) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_4)) / (Om / l))))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (U_42_ - U)))));
	} else {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * ((n * l) / Om))) * (((U - U_42_) / t_4) - (l * -2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = n * math.pow((l / Om), 2.0)
	t_3 = t_1 * ((t - (2.0 * ((l * l) / Om))) - (t_2 * (U - U_42_)))
	t_4 = Om / (n * l)
	tmp = 0
	if t_3 <= 0.0001:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_4)) / (Om / l))))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (U_42_ - U)))))
	else:
		tmp = math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * ((n * l) / Om))) * (((U - U_42_) / t_4) - (l * -2.0)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(t_2 * Float64(U - U_42_))))
	t_4 = Float64(Om / Float64(n * l))
	tmp = 0.0
	if (t_3 <= 0.0001)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(U_42_ - U) / t_4)) / Float64(Om / l))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + Float64(t_2 * Float64(U_42_ - U)))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) - Float64(Float64(2.0 * Float64(U * Float64(Float64(n * l) / Om))) * Float64(Float64(Float64(U - U_42_) / t_4) - Float64(l * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U * (2.0 * n);
	t_2 = n * ((l / Om) ^ 2.0);
	t_3 = t_1 * ((t - (2.0 * ((l * l) / Om))) - (t_2 * (U - U_42_)));
	t_4 = Om / (n * l);
	tmp = 0.0;
	if (t_3 <= 0.0001)
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_4)) / (Om / l))))));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (U_42_ - U)))));
	else
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * ((n * l) / Om))) * (((U - U_42_) / t_4) - (l * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0001], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(U * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / t$95$4), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000005e-4

   1. Initial program 58.3%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 63.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 74.5%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 68.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 73.7%

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

   4. Applied egg-rr 73.7%

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

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

   1. Initial program 0.0%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 25.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 66.7%

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

   9. Simplified 66.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0.0001:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{\frac{Om}{n \cdot \ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_1 \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot t\_1\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}^{0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* n l) Om)))
   (if (<= n -4.5e-10)
     (sqrt
      (* (* 2.0 (* n U)) (+ t (* (/ l Om) (+ (* l -2.0) (* t_1 (- U* U)))))))
     (if (<= n 8.8e-17)
       (sqrt
        (-
         (* U (* t (* 2.0 n)))
         (* (* 2.0 (* U t_1)) (- (/ (- U U*) (/ Om (* n l))) (* l -2.0)))))
       (*
        (pow (* U (+ t (/ (/ U* (/ (/ Om l) n)) (/ Om l)))) 0.5)
        (pow (* 2.0 n) 0.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double tmp;
	if (n <= -4.5e-10) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (n <= 8.8e-17) {
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	} else {
		tmp = pow((U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l)))), 0.5) * pow((2.0 * n), 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) :: t_1
    real(8) :: tmp
    t_1 = (n * l) / om
    if (n <= (-4.5d-10)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * ((l * (-2.0d0)) + (t_1 * (u_42 - u)))))))
    else if (n <= 8.8d-17) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) - ((2.0d0 * (u * t_1)) * (((u - u_42) / (om / (n * l))) - (l * (-2.0d0))))))
    else
        tmp = ((u * (t + ((u_42 / ((om / l) / n)) / (om / l)))) ** 0.5d0) * ((2.0d0 * n) ** 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 t_1 = (n * l) / Om;
	double tmp;
	if (n <= -4.5e-10) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (n <= 8.8e-17) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	} else {
		tmp = Math.pow((U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l)))), 0.5) * Math.pow((2.0 * n), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * l) / Om
	tmp = 0
	if n <= -4.5e-10:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))))
	elif n <= 8.8e-17:
		tmp = math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))))
	else:
		tmp = math.pow((U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l)))), 0.5) * math.pow((2.0 * n), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * l) / Om)
	tmp = 0.0
	if (n <= -4.5e-10)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(t_1 * Float64(U_42_ - U)))))));
	elseif (n <= 8.8e-17)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) - Float64(Float64(2.0 * Float64(U * t_1)) * Float64(Float64(Float64(U - U_42_) / Float64(Om / Float64(n * l))) - Float64(l * -2.0)))));
	else
		tmp = Float64((Float64(U * Float64(t + Float64(Float64(U_42_ / Float64(Float64(Om / l) / n)) / Float64(Om / l)))) ^ 0.5) * (Float64(2.0 * n) ^ 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * l) / Om;
	tmp = 0.0;
	if (n <= -4.5e-10)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	elseif (n <= 8.8e-17)
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	else
		tmp = ((U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l)))) ^ 0.5) * ((2.0 * n) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[n, -4.5e-10], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 8.8e-17], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(U * N[(t + N[(N[(U$42$ / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.5e-10

   1. Initial program 50.1%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 65.4%

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

   if -4.5e-10 < n < 8.8e-17

   1. Initial program 49.6%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 56.1%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 67.2%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 73.6%

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

   9. Simplified 73.6%

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

   if 8.8e-17 < n

   1. Initial program 66.6%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 69.1%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 64.6%

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

   7. Simplified 64.6%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow-prod-down N/A

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

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

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}^{0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;\ell \leq 8.6 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_1 \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot t\_1\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* n l) Om)))
   (if (<= l 8.6e-16)
     (sqrt
      (* (* 2.0 (* n U)) (+ t (* (/ l Om) (+ (* l -2.0) (* t_1 (- U* U)))))))
     (if (<= l 6.8e+249)
       (sqrt
        (-
         (* U (* t (* 2.0 n)))
         (* (* 2.0 (* U t_1)) (- (/ (- U U*) (/ Om (* n l))) (* l -2.0)))))
       (*
        (sqrt (/ (* (* n U) (+ -2.0 (/ (* n (- U* U)) Om))) Om))
        (* l (sqrt 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double tmp;
	if (l <= 8.6e-16) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (l <= 6.8e+249) {
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	} else {
		tmp = sqrt((((n * U) * (-2.0 + ((n * (U_42_ - U)) / Om))) / Om)) * (l * sqrt(2.0));
	}
	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 = (n * l) / om
    if (l <= 8.6d-16) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * ((l * (-2.0d0)) + (t_1 * (u_42 - u)))))))
    else if (l <= 6.8d+249) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) - ((2.0d0 * (u * t_1)) * (((u - u_42) / (om / (n * l))) - (l * (-2.0d0))))))
    else
        tmp = sqrt((((n * u) * ((-2.0d0) + ((n * (u_42 - u)) / om))) / om)) * (l * sqrt(2.0d0))
    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 = (n * l) / Om;
	double tmp;
	if (l <= 8.6e-16) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (l <= 6.8e+249) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	} else {
		tmp = Math.sqrt((((n * U) * (-2.0 + ((n * (U_42_ - U)) / Om))) / Om)) * (l * Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * l) / Om
	tmp = 0
	if l <= 8.6e-16:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))))
	elif l <= 6.8e+249:
		tmp = math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))))
	else:
		tmp = math.sqrt((((n * U) * (-2.0 + ((n * (U_42_ - U)) / Om))) / Om)) * (l * math.sqrt(2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * l) / Om)
	tmp = 0.0
	if (l <= 8.6e-16)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(t_1 * Float64(U_42_ - U)))))));
	elseif (l <= 6.8e+249)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) - Float64(Float64(2.0 * Float64(U * t_1)) * Float64(Float64(Float64(U - U_42_) / Float64(Om / Float64(n * l))) - Float64(l * -2.0)))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(n * U) * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Om)) * Float64(l * sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * l) / Om;
	tmp = 0.0;
	if (l <= 8.6e-16)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	elseif (l <= 6.8e+249)
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	else
		tmp = sqrt((((n * U) * (-2.0 + ((n * (U_42_ - U)) / Om))) / Om)) * (l * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, 8.6e-16], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.8e+249], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision] * N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.5999999999999997e-16

   1. Initial program 57.3%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 64.0%

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

   if 8.5999999999999997e-16 < l < 6.80000000000000026e249

   1. Initial program 42.8%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 53.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 53.1%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 55.3%

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

   9. Simplified 55.3%

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

   if 6.80000000000000026e249 < l

   1. Initial program 31.1%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 48.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

   7. Simplified 49.5%

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

   8. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_1 \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot t\_1\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* n l) Om)))
   (if (<= l 3.1e-16)
     (sqrt
      (* (* 2.0 (* n U)) (+ t (* (/ l Om) (+ (* l -2.0) (* t_1 (- U* U)))))))
     (if (<= l 1.1e+250)
       (sqrt
        (-
         (* U (* t (* 2.0 n)))
         (* (* 2.0 (* U t_1)) (- (/ (- U U*) (/ Om (* n l))) (* l -2.0)))))
       (*
        (* l (sqrt 2.0))
        (sqrt (/ (* (* n U) (+ -2.0 (/ (* n U*) Om))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double tmp;
	if (l <= 3.1e-16) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (l <= 1.1e+250) {
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((((n * U) * (-2.0 + ((n * U_42_) / 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) :: t_1
    real(8) :: tmp
    t_1 = (n * l) / om
    if (l <= 3.1d-16) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * ((l * (-2.0d0)) + (t_1 * (u_42 - u)))))))
    else if (l <= 1.1d+250) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) - ((2.0d0 * (u * t_1)) * (((u - u_42) / (om / (n * l))) - (l * (-2.0d0))))))
    else
        tmp = (l * sqrt(2.0d0)) * sqrt((((n * u) * ((-2.0d0) + ((n * u_42) / 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 t_1 = (n * l) / Om;
	double tmp;
	if (l <= 3.1e-16) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (l <= 1.1e+250) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	} else {
		tmp = (l * Math.sqrt(2.0)) * Math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * l) / Om
	tmp = 0
	if l <= 3.1e-16:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))))
	elif l <= 1.1e+250:
		tmp = math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))))
	else:
		tmp = (l * math.sqrt(2.0)) * math.sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * l) / Om)
	tmp = 0.0
	if (l <= 3.1e-16)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(t_1 * Float64(U_42_ - U)))))));
	elseif (l <= 1.1e+250)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) - Float64(Float64(2.0 * Float64(U * t_1)) * Float64(Float64(Float64(U - U_42_) / Float64(Om / Float64(n * l))) - Float64(l * -2.0)))));
	else
		tmp = Float64(Float64(l * sqrt(2.0)) * sqrt(Float64(Float64(Float64(n * U) * Float64(-2.0 + Float64(Float64(n * U_42_) / Om))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * l) / Om;
	tmp = 0.0;
	if (l <= 3.1e-16)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	elseif (l <= 1.1e+250)
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / (Om / (n * l))) - (l * -2.0)))));
	else
		tmp = (l * sqrt(2.0)) * sqrt((((n * U) * (-2.0 + ((n * U_42_) / Om))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l, 3.1e-16], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.1e+250], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * N[(-2.0 + N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.1000000000000001e-16

   1. Initial program 57.3%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 64.0%

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

   if 3.1000000000000001e-16 < l < 1.10000000000000007e250

   1. Initial program 42.8%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 53.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 53.1%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 55.3%

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

   9. Simplified 55.3%

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

   if 1.10000000000000007e250 < l

   1. Initial program 31.1%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 48.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

   7. Simplified 49.5%

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

   8. Taylor expanded in U around 0

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\left(n \cdot U\right) \cdot \left(-2 + \frac{n \cdot U*}{Om}\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n \cdot \ell}{Om}\\ t_2 := \frac{Om}{n \cdot \ell}\\ \mathbf{if}\;n \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_1 \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot t\_1\right)\right) \cdot \left(\frac{U - U*}{t\_2} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{t\_2}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* n l) Om)) (t_2 (/ Om (* n l))))
   (if (<= n -1.95e-5)
     (sqrt
      (* (* 2.0 (* n U)) (+ t (* (/ l Om) (+ (* l -2.0) (* t_1 (- U* U)))))))
     (if (<= n 2.8e-40)
       (sqrt
        (-
         (* U (* t (* 2.0 n)))
         (* (* 2.0 (* U t_1)) (- (/ (- U U*) t_2) (* l -2.0)))))
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (+ (* l -2.0) (/ (- U* U) t_2)) (/ Om l))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double t_2 = Om / (n * l);
	double tmp;
	if (n <= -1.95e-5) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (n <= 2.8e-40) {
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / t_2) - (l * -2.0)))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_2)) / (Om / l))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (n * l) / om
    t_2 = om / (n * l)
    if (n <= (-1.95d-5)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * ((l * (-2.0d0)) + (t_1 * (u_42 - u)))))))
    else if (n <= 2.8d-40) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) - ((2.0d0 * (u * t_1)) * (((u - u_42) / t_2) - (l * (-2.0d0))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l * (-2.0d0)) + ((u_42 - u) / t_2)) / (om / l))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double t_2 = Om / (n * l);
	double tmp;
	if (n <= -1.95e-5) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	} else if (n <= 2.8e-40) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / t_2) - (l * -2.0)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_2)) / (Om / l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * l) / Om
	t_2 = Om / (n * l)
	tmp = 0
	if n <= -1.95e-5:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))))
	elif n <= 2.8e-40:
		tmp = math.sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / t_2) - (l * -2.0)))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_2)) / (Om / l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * l) / Om)
	t_2 = Float64(Om / Float64(n * l))
	tmp = 0.0
	if (n <= -1.95e-5)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(t_1 * Float64(U_42_ - U)))))));
	elseif (n <= 2.8e-40)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) - Float64(Float64(2.0 * Float64(U * t_1)) * Float64(Float64(Float64(U - U_42_) / t_2) - Float64(l * -2.0)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(U_42_ - U) / t_2)) / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * l) / Om;
	t_2 = Om / (n * l);
	tmp = 0.0;
	if (n <= -1.95e-5)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	elseif (n <= 2.8e-40)
		tmp = sqrt(((U * (t * (2.0 * n))) - ((2.0 * (U * t_1)) * (((U - U_42_) / t_2) - (l * -2.0)))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / t_2)) / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.95e-5], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.8e-40], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.95e-5

   1. Initial program 50.0%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 65.8%

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

   if -1.95e-5 < n < 2.8e-40

   1. Initial program 50.0%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 55.9%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 66.5%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 73.1%

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

   9. Simplified 73.1%

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

   if 2.8e-40 < n

   1. Initial program 64.5%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 68.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 70.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) - \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right) \cdot \left(\frac{U - U*}{\frac{Om}{n \cdot \ell}} - \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{\frac{Om}{n \cdot \ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n \cdot \ell}{Om}\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;U* \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq -1.12 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right) - \ell \cdot -2\right)\right)}\\ \mathbf{elif}\;U* \leq 1.05 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(2 \cdot \left(U \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_1 \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* n l) Om)) (t_2 (* 2.0 (* n U))))
   (if (<= U* -4.5e+88)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (/ U* (/ (/ Om l) n)) (/ Om l))))))
     (if (<= U* -1.12e-282)
       (sqrt
        (* t_2 (- t (* (/ l Om) (- (* (/ l Om) (* n (- U U*))) (* l -2.0))))))
       (if (<= U* 1.05e-56)
         (sqrt (+ (* U (* t (* 2.0 n))) (* (* l -2.0) (* 2.0 (* U t_1)))))
         (sqrt (* t_2 (+ t (* (/ l Om) (+ (* l -2.0) (* t_1 (- U* U))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (U_42_ <= -4.5e+88) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= -1.12e-282) {
		tmp = sqrt((t_2 * (t - ((l / Om) * (((l / Om) * (n * (U - U_42_))) - (l * -2.0))))));
	} else if (U_42_ <= 1.05e-56) {
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))));
	} else {
		tmp = sqrt((t_2 * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (n * l) / om
    t_2 = 2.0d0 * (n * u)
    if (u_42 <= (-4.5d+88)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((u_42 / ((om / l) / n)) / (om / l))))))
    else if (u_42 <= (-1.12d-282)) then
        tmp = sqrt((t_2 * (t - ((l / om) * (((l / om) * (n * (u - u_42))) - (l * (-2.0d0)))))))
    else if (u_42 <= 1.05d-56) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) + ((l * (-2.0d0)) * (2.0d0 * (u * t_1)))))
    else
        tmp = sqrt((t_2 * (t + ((l / om) * ((l * (-2.0d0)) + (t_1 * (u_42 - 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 t_1 = (n * l) / Om;
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (U_42_ <= -4.5e+88) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= -1.12e-282) {
		tmp = Math.sqrt((t_2 * (t - ((l / Om) * (((l / Om) * (n * (U - U_42_))) - (l * -2.0))))));
	} else if (U_42_ <= 1.05e-56) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))));
	} else {
		tmp = Math.sqrt((t_2 * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * l) / Om
	t_2 = 2.0 * (n * U)
	tmp = 0
	if U_42_ <= -4.5e+88:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))))
	elif U_42_ <= -1.12e-282:
		tmp = math.sqrt((t_2 * (t - ((l / Om) * (((l / Om) * (n * (U - U_42_))) - (l * -2.0))))))
	elif U_42_ <= 1.05e-56:
		tmp = math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))))
	else:
		tmp = math.sqrt((t_2 * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * l) / Om)
	t_2 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (U_42_ <= -4.5e+88)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(U_42_ / Float64(Float64(Om / l) / n)) / Float64(Om / l))))));
	elseif (U_42_ <= -1.12e-282)
		tmp = sqrt(Float64(t_2 * Float64(t - Float64(Float64(l / Om) * Float64(Float64(Float64(l / Om) * Float64(n * Float64(U - U_42_))) - Float64(l * -2.0))))));
	elseif (U_42_ <= 1.05e-56)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) + Float64(Float64(l * -2.0) * Float64(2.0 * Float64(U * t_1)))));
	else
		tmp = sqrt(Float64(t_2 * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(t_1 * Float64(U_42_ - U)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * l) / Om;
	t_2 = 2.0 * (n * U);
	tmp = 0.0;
	if (U_42_ <= -4.5e+88)
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	elseif (U_42_ <= -1.12e-282)
		tmp = sqrt((t_2 * (t - ((l / Om) * (((l / Om) * (n * (U - U_42_))) - (l * -2.0))))));
	elseif (U_42_ <= 1.05e-56)
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))));
	else
		tmp = sqrt((t_2 * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U$42$, -4.5e+88], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, -1.12e-282], N[Sqrt[N[(t$95$2 * N[(t - N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 1.05e-56], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * -2.0), $MachinePrecision] * N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if U* < -4.5e88

   1. Initial program 60.8%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 64.6%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 63.0%

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

   7. Simplified 63.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

   9. Applied egg-rr 67.6%

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

   if -4.5e88 < U* < -1.11999999999999992e-282

   1. Initial program 51.2%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 58.8%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 63.3%

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

   6. Applied egg-rr 63.3%

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

   if -1.11999999999999992e-282 < U* < 1.05000000000000003e-56

   1. Initial program 46.0%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 52.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 67.7%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in Om around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 72.9%

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

   12. Simplified 72.9%

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

   if 1.05000000000000003e-56 < U*

   1. Initial program 56.7%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 69.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq -1.12 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right) - \ell \cdot -2\right)\right)}\\ \mathbf{elif}\;U* \leq 1.05 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;U* \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq -3.5 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(2 \cdot \left(U \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot t\_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))) (t_2 (/ (* n l) Om)))
   (if (<= U* -1.05e-21)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (/ U* (/ (/ Om l) n)) (/ Om l))))))
     (if (<= U* -3.5e-282)
       (sqrt (* t_1 (+ t (* (/ l Om) (* l -2.0)))))
       (if (<= U* 1.65e-50)
         (sqrt (+ (* U (* t (* 2.0 n))) (* (* l -2.0) (* 2.0 (* U t_2)))))
         (sqrt (* t_1 (+ t (* (/ l Om) (* U* t_2))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = (n * l) / Om;
	double tmp;
	if (U_42_ <= -1.05e-21) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= -3.5e-282) {
		tmp = sqrt((t_1 * (t + ((l / Om) * (l * -2.0)))));
	} else if (U_42_ <= 1.65e-50) {
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_2)))));
	} else {
		tmp = sqrt((t_1 * (t + ((l / Om) * (U_42_ * t_2)))));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    t_2 = (n * l) / om
    if (u_42 <= (-1.05d-21)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((u_42 / ((om / l) / n)) / (om / l))))))
    else if (u_42 <= (-3.5d-282)) then
        tmp = sqrt((t_1 * (t + ((l / om) * (l * (-2.0d0))))))
    else if (u_42 <= 1.65d-50) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) + ((l * (-2.0d0)) * (2.0d0 * (u * t_2)))))
    else
        tmp = sqrt((t_1 * (t + ((l / om) * (u_42 * t_2)))))
    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 * (n * U);
	double t_2 = (n * l) / Om;
	double tmp;
	if (U_42_ <= -1.05e-21) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= -3.5e-282) {
		tmp = Math.sqrt((t_1 * (t + ((l / Om) * (l * -2.0)))));
	} else if (U_42_ <= 1.65e-50) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_2)))));
	} else {
		tmp = Math.sqrt((t_1 * (t + ((l / Om) * (U_42_ * t_2)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	t_2 = (n * l) / Om
	tmp = 0
	if U_42_ <= -1.05e-21:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))))
	elif U_42_ <= -3.5e-282:
		tmp = math.sqrt((t_1 * (t + ((l / Om) * (l * -2.0)))))
	elif U_42_ <= 1.65e-50:
		tmp = math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_2)))))
	else:
		tmp = math.sqrt((t_1 * (t + ((l / Om) * (U_42_ * t_2)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(Float64(n * l) / Om)
	tmp = 0.0
	if (U_42_ <= -1.05e-21)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(U_42_ / Float64(Float64(Om / l) / n)) / Float64(Om / l))))));
	elseif (U_42_ <= -3.5e-282)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l / Om) * Float64(l * -2.0)))));
	elseif (U_42_ <= 1.65e-50)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) + Float64(Float64(l * -2.0) * Float64(2.0 * Float64(U * t_2)))));
	else
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	t_2 = (n * l) / Om;
	tmp = 0.0;
	if (U_42_ <= -1.05e-21)
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	elseif (U_42_ <= -3.5e-282)
		tmp = sqrt((t_1 * (t + ((l / Om) * (l * -2.0)))));
	elseif (U_42_ <= 1.65e-50)
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_2)))));
	else
		tmp = sqrt((t_1 * (t + ((l / Om) * (U_42_ * t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[U$42$, -1.05e-21], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, -3.5e-282], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 1.65e-50], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * -2.0), $MachinePrecision] * N[(2.0 * N[(U * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if U* < -1.05000000000000006e-21

   1. Initial program 57.1%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 62.6%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 59.6%

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

   7. Simplified 59.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

   9. Applied egg-rr 62.8%

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

   if -1.05000000000000006e-21 < U* < -3.50000000000000006e-282

   1. Initial program 52.4%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 58.9%

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

   5. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 61.0%

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

   7. Simplified 61.0%

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

   if -3.50000000000000006e-282 < U* < 1.6499999999999999e-50

   1. Initial program 46.0%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 52.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

   6. Applied egg-rr 67.7%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in Om around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 72.9%

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

   12. Simplified 72.9%

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

   if 1.6499999999999999e-50 < U*

   1. Initial program 56.7%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 69.1%

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

   5. Taylor expanded in U* around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. *-lowering-*.f64 62.2%

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

   7. Simplified 62.2%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq -3.5 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{n \cdot \ell}{Om}\\ \mathbf{if}\;U* \leq -6.2 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{\frac{Om}{n \cdot \ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(2 \cdot \left(U \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + t\_1 \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* n l) Om)))
   (if (<= U* -6.2e-141)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (/ (+ (* l -2.0) (/ (- U* U) (/ Om (* n l)))) (/ Om l))))))
     (if (<= U* 2.9e-58)
       (sqrt (+ (* U (* t (* 2.0 n))) (* (* l -2.0) (* 2.0 (* U t_1)))))
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (* (/ l Om) (+ (* l -2.0) (* t_1 (- U* U)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * l) / Om;
	double tmp;
	if (U_42_ <= -6.2e-141) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / (Om / (n * l)))) / (Om / l))))));
	} else if (U_42_ <= 2.9e-58) {
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - 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) :: t_1
    real(8) :: tmp
    t_1 = (n * l) / om
    if (u_42 <= (-6.2d-141)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l * (-2.0d0)) + ((u_42 - u) / (om / (n * l)))) / (om / l))))))
    else if (u_42 <= 2.9d-58) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) + ((l * (-2.0d0)) * (2.0d0 * (u * t_1)))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * ((l * (-2.0d0)) + (t_1 * (u_42 - 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 t_1 = (n * l) / Om;
	double tmp;
	if (U_42_ <= -6.2e-141) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / (Om / (n * l)))) / (Om / l))))));
	} else if (U_42_ <= 2.9e-58) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * l) / Om
	tmp = 0
	if U_42_ <= -6.2e-141:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / (Om / (n * l)))) / (Om / l))))))
	elif U_42_ <= 2.9e-58:
		tmp = math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * l) / Om)
	tmp = 0.0
	if (U_42_ <= -6.2e-141)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) + Float64(Float64(U_42_ - U) / Float64(Om / Float64(n * l)))) / Float64(Om / l))))));
	elseif (U_42_ <= 2.9e-58)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) + Float64(Float64(l * -2.0) * Float64(2.0 * Float64(U * t_1)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(l * -2.0) + Float64(t_1 * Float64(U_42_ - U)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * l) / Om;
	tmp = 0.0;
	if (U_42_ <= -6.2e-141)
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) + ((U_42_ - U) / (Om / (n * l)))) / (Om / l))))));
	elseif (U_42_ <= 2.9e-58)
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * (2.0 * (U * t_1)))));
	else
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * ((l * -2.0) + (t_1 * (U_42_ - U)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[U$42$, -6.2e-141], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(U$42$ - U), $MachinePrecision] / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 2.9e-58], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * -2.0), $MachinePrecision] * N[(2.0 * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if U* < -6.20000000000000055e-141

   1. Initial program 56.8%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

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

   3. Simplified 63.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 66.4%

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

   if -6.20000000000000055e-141 < U* < 2.8999999999999999e-58

   1. Initial program 47.2%

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

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

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

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

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right) + t\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) + \left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right) + \left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right), \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)\right) \]

   6. Applied egg-rr 61.5%

      \[\leadsto \sqrt{\color{blue}{\left(\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) + U \cdot \left(\left(n \cdot 2\right) \cdot t\right)}} \]

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(n, \ell\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot n\right)}{Om}\right)}\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(n, \ell\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{*.f64}\left(2, \left(\frac{U \cdot \left(\ell \cdot n\right)}{Om}\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(n, \ell\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{*.f64}\left(2, \left(U \cdot \frac{\ell \cdot n}{Om}\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(n, \ell\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(n, \ell\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

      5. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(n, \ell\right)\right)\right), \mathsf{*.f64}\left(\ell, -2\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

   9. Simplified 66.2%

      \[\leadsto \sqrt{\left(\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2\right) \cdot \color{blue}{\left(2 \cdot \left(U \cdot \frac{\ell \cdot n}{Om}\right)\right)} + U \cdot \left(\left(n \cdot 2\right) \cdot t\right)} \]

   10. Taylor expanded in Om around inf

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot -2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

       2. *-lowering-*.f64 66.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

   12. Simplified 66.6%

       \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot -2\right)} \cdot \left(2 \cdot \left(U \cdot \frac{\ell \cdot n}{Om}\right)\right) + U \cdot \left(\left(n \cdot 2\right) \cdot t\right)} \]

   if 2.8999999999999999e-58 < U*

   1. Initial program 56.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -6.2 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 + \frac{U* - U}{\frac{Om}{n \cdot \ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(2 \cdot \left(U \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(\left(2 \cdot U\right) \cdot \left(\ell \cdot \left(n \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= l 4e-68)
     (sqrt (* t_1 (+ t (* (/ l Om) (* U* (/ (* n l) Om))))))
     (if (<= l 3.1e+167)
       (sqrt (* t_1 (+ t (* (/ l Om) (* l (+ -2.0 (* n (/ (- U* U) Om))))))))
       (sqrt
        (/
         (* l (* (* 2.0 U) (* l (* n (+ -2.0 (/ (* n (- U* U)) Om))))))
         Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 4e-68) {
		tmp = sqrt((t_1 * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	} else if (l <= 3.1e+167) {
		tmp = sqrt((t_1 * (t + ((l / Om) * (l * (-2.0 + (n * ((U_42_ - U) / Om))))))));
	} else {
		tmp = sqrt(((l * ((2.0 * U) * (l * (n * (-2.0 + ((n * (U_42_ - U)) / 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (l <= 4d-68) then
        tmp = sqrt((t_1 * (t + ((l / om) * (u_42 * ((n * l) / om))))))
    else if (l <= 3.1d+167) then
        tmp = sqrt((t_1 * (t + ((l / om) * (l * ((-2.0d0) + (n * ((u_42 - u) / om))))))))
    else
        tmp = sqrt(((l * ((2.0d0 * u) * (l * (n * ((-2.0d0) + ((n * (u_42 - u)) / 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 t_1 = 2.0 * (n * U);
	double tmp;
	if (l <= 4e-68) {
		tmp = Math.sqrt((t_1 * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	} else if (l <= 3.1e+167) {
		tmp = Math.sqrt((t_1 * (t + ((l / Om) * (l * (-2.0 + (n * ((U_42_ - U) / Om))))))));
	} else {
		tmp = Math.sqrt(((l * ((2.0 * U) * (l * (n * (-2.0 + ((n * (U_42_ - U)) / Om)))))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if l <= 4e-68:
		tmp = math.sqrt((t_1 * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))))
	elif l <= 3.1e+167:
		tmp = math.sqrt((t_1 * (t + ((l / Om) * (l * (-2.0 + (n * ((U_42_ - U) / Om))))))))
	else:
		tmp = math.sqrt(((l * ((2.0 * U) * (l * (n * (-2.0 + ((n * (U_42_ - U)) / Om)))))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 4e-68)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om))))));
	elseif (l <= 3.1e+167)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(l / Om) * Float64(l * Float64(-2.0 + Float64(n * Float64(Float64(U_42_ - U) / Om))))))));
	else
		tmp = sqrt(Float64(Float64(l * Float64(Float64(2.0 * U) * Float64(l * Float64(n * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om)))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 4e-68)
		tmp = sqrt((t_1 * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	elseif (l <= 3.1e+167)
		tmp = sqrt((t_1 * (t + ((l / Om) * (l * (-2.0 + (n * ((U_42_ - U) / Om))))))));
	else
		tmp = sqrt(((l * ((2.0 * U) * (l * (n * (-2.0 + ((n * (U_42_ - U)) / Om)))))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4e-68], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.1e+167], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * N[(-2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(l * N[(N[(2.0 * U), $MachinePrecision] * N[(l * N[(n * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.00000000000000027e-68

   1. Initial program 57.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 59.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 59.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)} \]

   if 4.00000000000000027e-68 < l < 3.1e167

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om} + -2 \cdot \ell\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \frac{n \cdot \left(U* - U\right)}{Om} + -2 \cdot \ell\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \frac{n \cdot \left(U* - U\right)}{Om} + \ell \cdot -2\right)\right)\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} + -2\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(\frac{n \cdot \left(U* - U\right)}{Om} + -2\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(\frac{n \cdot \left(U* - U\right)}{Om}\right), -2\right)\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(n \cdot \frac{U* - U}{Om}\right), -2\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{U* - U}{Om}\right)\right), -2\right)\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\left(U* - U\right), Om\right)\right), -2\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 62.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(U*, U\right), Om\right)\right), -2\right)\right)\right)\right)\right)\right) \]

   7. Simplified 62.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)}\right)} \]

   if 3.1e167 < l

   1. Initial program 25.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   7. Simplified 41.6%

      \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)\right)\right)\right)}{Om}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(\ell \cdot \left(n \cdot \left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)\right)\right)\right), Om\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(\left(n \cdot \left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)\right) \cdot \ell\right)\right), Om\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)\right)\right) \cdot \ell\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)\right)\right), \ell\right), Om\right)\right) \]

   9. Applied egg-rr 55.4%

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(U \cdot 2\right) \cdot \left(\ell \cdot \left(\left(-2 + \frac{\left(U* - U\right) \cdot n}{Om}\right) \cdot n\right)\right)\right) \cdot \ell}}{Om}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(\left(2 \cdot U\right) \cdot \left(\ell \cdot \left(n \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -7.2 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -7.2e-139)
   (sqrt (* (* 2.0 n) (* U (+ t (/ (/ U* (/ (/ Om l) n)) (/ Om l))))))
   (if (<= U* 1.65e-50)
     (sqrt
      (+ (* U (* t (* 2.0 n))) (* (* l -2.0) (* (* U (* 2.0 n)) (/ l Om)))))
     (sqrt (* (* 2.0 (* n U)) (+ t (* (/ l Om) (* U* (/ (* n l) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -7.2e-139) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= 1.65e-50) {
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * ((U * (2.0 * n)) * (l / Om)))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((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 (u_42 <= (-7.2d-139)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((u_42 / ((om / l) / n)) / (om / l))))))
    else if (u_42 <= 1.65d-50) then
        tmp = sqrt(((u * (t * (2.0d0 * n))) + ((l * (-2.0d0)) * ((u * (2.0d0 * n)) * (l / om)))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * (u_42 * ((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 (U_42_ <= -7.2e-139) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= 1.65e-50) {
		tmp = Math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * ((U * (2.0 * n)) * (l / Om)))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -7.2e-139:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))))
	elif U_42_ <= 1.65e-50:
		tmp = math.sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * ((U * (2.0 * n)) * (l / Om)))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -7.2e-139)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(U_42_ / Float64(Float64(Om / l) / n)) / Float64(Om / l))))));
	elseif (U_42_ <= 1.65e-50)
		tmp = sqrt(Float64(Float64(U * Float64(t * Float64(2.0 * n))) + Float64(Float64(l * -2.0) * Float64(Float64(U * Float64(2.0 * n)) * Float64(l / Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -7.2e-139)
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	elseif (U_42_ <= 1.65e-50)
		tmp = sqrt(((U * (t * (2.0 * n))) + ((l * -2.0) * ((U * (2.0 * n)) * (l / Om)))));
	else
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -7.2e-139], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 1.65e-50], N[Sqrt[N[(N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * -2.0), $MachinePrecision] * N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U* < -7.20000000000000007e-139

   1. Initial program 56.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 58.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 58.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot U\right) \cdot \left(n \cdot 2\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot U\right), \left(n \cdot 2\right)\right)\right) \]

   9. Applied egg-rr 61.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   if -7.20000000000000007e-139 < U* < 1.6499999999999999e-50

   1. Initial program 47.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right) + t\right)\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) + \left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right) + \left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right), \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)\right) \]

   6. Applied egg-rr 61.5%

      \[\leadsto \sqrt{\color{blue}{\left(\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) + U \cdot \left(\left(n \cdot 2\right) \cdot t\right)}} \]

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, 2\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, 2\right)\right)\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, 2\right), t\right)\right)\right)\right) \]

   9. Simplified 64.1%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot \ell\right)} \cdot \left(\frac{\ell}{Om} \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) + U \cdot \left(\left(n \cdot 2\right) \cdot t\right)} \]

   if 1.6499999999999999e-50 < U*

   1. Initial program 56.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 62.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 62.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -7.2 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 1.65 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(2 \cdot n\right)\right) + \left(\ell \cdot -2\right) \cdot \left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 4.8 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* -7.5e-13)
   (sqrt (* (* 2.0 n) (* U (+ t (/ (/ U* (/ (/ Om l) n)) (/ Om l))))))
   (if (<= U* 4.8e+91)
     (sqrt (* (* 2.0 n) (* U (+ t (/ (* l -2.0) (/ Om l))))))
     (sqrt (* (* 2.0 (* n U)) (+ t (* (/ l Om) (* U* (/ (* n l) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= -7.5e-13) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= 4.8e+91) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((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 (u_42 <= (-7.5d-13)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((u_42 / ((om / l) / n)) / (om / l))))))
    else if (u_42 <= 4.8d+91) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * (-2.0d0)) / (om / l))))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * (u_42 * ((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 (U_42_ <= -7.5e-13) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	} else if (U_42_ <= 4.8e+91) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= -7.5e-13:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))))
	elif U_42_ <= 4.8e+91:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= -7.5e-13)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(U_42_ / Float64(Float64(Om / l) / n)) / Float64(Om / l))))));
	elseif (U_42_ <= 4.8e+91)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= -7.5e-13)
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ / ((Om / l) / n)) / (Om / l))))));
	elseif (U_42_ <= 4.8e+91)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	else
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, -7.5e-13], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ / N[(N[(Om / l), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U$42$, 4.8e+91], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U* < -7.5000000000000004e-13

   1. Initial program 58.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 60.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 60.9%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot U\right) \cdot \left(n \cdot 2\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right) \cdot U\right), \left(n \cdot 2\right)\right)\right) \]

   9. Applied egg-rr 63.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   if -7.5000000000000004e-13 < U* < 4.79999999999999966e91

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 58.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), U\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 58.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), U\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]

   9. Simplified 58.4%

      \[\leadsto \sqrt{\left(\left(t + \frac{\color{blue}{-2 \cdot \ell}}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)} \]

   if 4.79999999999999966e91 < U*

   1. Initial program 53.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 64.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 64.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U*}{\frac{\frac{Om}{\ell}}{n}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;U* \leq 4.8 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{if}\;Om \leq -3 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 3.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* 2.0 n) (* U (+ t (/ (* l -2.0) (/ Om l))))))))
   (if (<= Om -3e+110)
     t_1
     (if (<= Om 3.8e+103)
       (sqrt (* (* 2.0 (* n U)) (+ t (* (/ l Om) (* U* (/ (* n l) Om))))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	double tmp;
	if (Om <= -3e+110) {
		tmp = t_1;
	} else if (Om <= 3.8e+103) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	} else {
		tmp = 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 = sqrt(((2.0d0 * n) * (u * (t + ((l * (-2.0d0)) / (om / l))))))
    if (om <= (-3d+110)) then
        tmp = t_1
    else if (om <= 3.8d+103) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * (u_42 * ((n * l) / om))))))
    else
        tmp = 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 = Math.sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	double tmp;
	if (Om <= -3e+110) {
		tmp = t_1;
	} else if (Om <= 3.8e+103) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))))
	tmp = 0
	if Om <= -3e+110:
		tmp = t_1
	elif Om <= 3.8e+103:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))))
	tmp = 0.0
	if (Om <= -3e+110)
		tmp = t_1;
	elseif (Om <= 3.8e+103)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(U_42_ * Float64(Float64(n * l) / Om))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	tmp = 0.0;
	if (Om <= -3e+110)
		tmp = t_1;
	elseif (Om <= 3.8e+103)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (U_42_ * ((n * l) / Om))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -3e+110], t$95$1, If[LessEqual[Om, 3.8e+103], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(U$42$ * N[(N[(n * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -3.00000000000000007e110 or 3.7999999999999997e103 < Om

   1. Initial program 56.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 66.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), U\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 67.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), U\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]

   9. Simplified 67.2%

      \[\leadsto \sqrt{\left(\left(t + \frac{\color{blue}{-2 \cdot \ell}}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)} \]

   if -3.00000000000000007e110 < Om < 3.7999999999999997e103

   1. Initial program 51.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in U* around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(U* \cdot \frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \left(\frac{\ell \cdot n}{Om}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\left(\ell \cdot n\right), Om\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(U*, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, n\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 56.9%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(U* \cdot \frac{\ell \cdot n}{Om}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -3 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;Om \leq 3.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(U* \cdot \frac{n \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \mathbf{if}\;U \leq -1.5 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* 2.0 (* n U)) (+ t (* (/ l Om) (* l -2.0)))))))
   (if (<= U -1.5e-43)
     t_1
     (if (<= U 1.32e+118)
       (sqrt (* (* 2.0 n) (* U (+ t (/ (* l -2.0) (/ Om l))))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))));
	double tmp;
	if (U <= -1.5e-43) {
		tmp = t_1;
	} else if (U <= 1.32e+118) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	} else {
		tmp = 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 = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * (l * (-2.0d0))))))
    if (u <= (-1.5d-43)) then
        tmp = t_1
    else if (u <= 1.32d+118) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * (-2.0d0)) / (om / l))))))
    else
        tmp = 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 = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))));
	double tmp;
	if (U <= -1.5e-43) {
		tmp = t_1;
	} else if (U <= 1.32e+118) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))))
	tmp = 0
	if U <= -1.5e-43:
		tmp = t_1
	elif U <= 1.32e+118:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(l * -2.0)))))
	tmp = 0.0
	if (U <= -1.5e-43)
		tmp = t_1;
	elseif (U <= 1.32e+118)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))));
	tmp = 0.0;
	if (U <= -1.5e-43)
		tmp = t_1;
	elseif (U <= 1.32e+118)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * -2.0) / (Om / l))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -1.5e-43], t$95$1, If[LessEqual[U, 1.32e+118], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U < -1.50000000000000002e-43 or 1.3199999999999999e118 < U

   1. Initial program 68.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 64.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 64.7%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)} \]

   if -1.50000000000000002e-43 < U < 1.3199999999999999e118

   1. Initial program 45.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 61.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   7. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), U\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 48.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), U\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]

   9. Simplified 48.3%

      \[\leadsto \sqrt{\left(\left(t + \frac{\color{blue}{-2 \cdot \ell}}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.5 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (* U (* t (* 2.0 n))) 0.5)))
   (if (<= t -4e-136)
     t_1
     (if (<= t 8.6e-292)
       (sqrt (/ (* -4.0 (* U (* n (* l l)))) Om))
       (if (<= t 5.8e+215) (sqrt (* (* U (* 2.0 n)) t)) t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((U * (t * (2.0 * n))), 0.5);
	double tmp;
	if (t <= -4e-136) {
		tmp = t_1;
	} else if (t <= 8.6e-292) {
		tmp = sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	} else if (t <= 5.8e+215) {
		tmp = sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = 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 = (u * (t * (2.0d0 * n))) ** 0.5d0
    if (t <= (-4d-136)) then
        tmp = t_1
    else if (t <= 8.6d-292) then
        tmp = sqrt((((-4.0d0) * (u * (n * (l * l)))) / om))
    else if (t <= 5.8d+215) then
        tmp = sqrt(((u * (2.0d0 * n)) * t))
    else
        tmp = 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 = Math.pow((U * (t * (2.0 * n))), 0.5);
	double tmp;
	if (t <= -4e-136) {
		tmp = t_1;
	} else if (t <= 8.6e-292) {
		tmp = Math.sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	} else if (t <= 5.8e+215) {
		tmp = Math.sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((U * (t * (2.0 * n))), 0.5)
	tmp = 0
	if t <= -4e-136:
		tmp = t_1
	elif t <= 8.6e-292:
		tmp = math.sqrt(((-4.0 * (U * (n * (l * l)))) / Om))
	elif t <= 5.8e+215:
		tmp = math.sqrt(((U * (2.0 * n)) * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(t * Float64(2.0 * n))) ^ 0.5
	tmp = 0.0
	if (t <= -4e-136)
		tmp = t_1;
	elseif (t <= 8.6e-292)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(U * Float64(n * Float64(l * l)))) / Om));
	elseif (t <= 5.8e+215)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (U * (t * (2.0 * n))) ^ 0.5;
	tmp = 0.0;
	if (t <= -4e-136)
		tmp = t_1;
	elseif (t <= 8.6e-292)
		tmp = sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	elseif (t <= 5.8e+215)
		tmp = sqrt(((U * (2.0 * n)) * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[t, -4e-136], t$95$1, If[LessEqual[t, 8.6e-292], N[Sqrt[N[(N[(-4.0 * N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 5.8e+215], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.00000000000000001e-136 or 5.7999999999999999e215 < t

   1. Initial program 53.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 49.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 49.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(t \cdot \left(n \cdot 2\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(2 \cdot n\right) \cdot t\right)\right), \frac{1}{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \left(\left(2 \cdot n\right) \cdot t\right)\right), \frac{1}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \left(t \cdot \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \left(n \cdot 2\right)\right)\right), \frac{1}{2}\right) \]

      13. *-lowering-*.f64 54.8%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, 2\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 54.8%

      \[\leadsto \color{blue}{{\left(U \cdot \left(t \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}} \]

   if -4.00000000000000001e-136 < t < 8.6e-292

   1. Initial program 44.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \frac{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)\right)\right)}{Om}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)\right)\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell + \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}\right)\right)\right)\right)\right), Om\right)\right) \]

   7. Simplified 57.1%

      \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(\ell \cdot \left(n \cdot \left(\ell \cdot \left(n \cdot \frac{U* - U}{Om} + -2\right)\right)\right)\right)\right)}{Om}}} \]

   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. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 36.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right) \]

   10. Simplified 36.0%

       \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]

   if 8.6e-292 < t < 5.7999999999999999e215

   1. Initial program 56.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-136}:\\ \;\;\;\;{\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+67}:\\ \;\;\;\;{\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -2.9e+67)
   (pow (* U (* t (* 2.0 n))) 0.5)
   (sqrt (* (* 2.0 (* n U)) (+ t (* (/ l Om) (* l -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2.9e+67) {
		tmp = pow((U * (t * (2.0 * n))), 0.5);
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))));
	}
	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 <= (-2.9d+67)) then
        tmp = (u * (t * (2.0d0 * n))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((l / om) * (l * (-2.0d0))))))
    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 <= -2.9e+67) {
		tmp = Math.pow((U * (t * (2.0 * n))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -2.9e+67:
		tmp = math.pow((U * (t * (2.0 * n))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -2.9e+67)
		tmp = Float64(U * Float64(t * Float64(2.0 * n))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(l / Om) * Float64(l * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.9e+67)
		tmp = (U * (t * (2.0 * n))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * (n * U)) * (t + ((l / Om) * (l * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -2.9e+67], N[Power[N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.90000000000000023e67

   1. Initial program 55.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 61.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 61.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(t \cdot \left(n \cdot 2\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(2 \cdot n\right) \cdot t\right)\right), \frac{1}{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \left(\left(2 \cdot n\right) \cdot t\right)\right), \frac{1}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \left(t \cdot \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \left(n \cdot 2\right)\right)\right), \frac{1}{2}\right) \]

      13. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, 2\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{{\left(U \cdot \left(t \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}} \]

   if -2.90000000000000023e67 < t

   1. Initial program 52.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(-2 \cdot \ell\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 47.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(-2, \ell\right)\right)\right)\right)\right) \]

   7. Simplified 47.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \color{blue}{\left(-2 \cdot \ell\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+67}:\\ \;\;\;\;{\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e-123)
   (sqrt (* (* U (* 2.0 n)) 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 <= 4.5e-123) {
		tmp = sqrt(((U * (2.0 * n)) * 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 <= 4.5d-123) then
        tmp = sqrt(((u * (2.0d0 * n)) * 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 <= 4.5e-123) {
		tmp = Math.sqrt(((U * (2.0 * n)) * 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 <= 4.5e-123:
		tmp = math.sqrt(((U * (2.0 * n)) * 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 <= 4.5e-123)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * 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 <= 4.5e-123)
		tmp = sqrt(((U * (2.0 * n)) * 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, 4.5e-123], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $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 < 4.49999999999999993e-123

   1. Initial program 57.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 44.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if 4.49999999999999993e-123 < l

   1. Initial program 44.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   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. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      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 40.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right)\right) \]

   7. Simplified 40.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{if}\;U \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* U (* 2.0 n)) t))))
   (if (<= U -2e+16)
     t_1
     (if (<= U 1.3e+118) (sqrt (* n (* t (* 2.0 U)))) t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((U * (2.0 * n)) * t));
	double tmp;
	if (U <= -2e+16) {
		tmp = t_1;
	} else if (U <= 1.3e+118) {
		tmp = sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = 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 = sqrt(((u * (2.0d0 * n)) * t))
    if (u <= (-2d+16)) then
        tmp = t_1
    else if (u <= 1.3d+118) then
        tmp = sqrt((n * (t * (2.0d0 * u))))
    else
        tmp = 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 = Math.sqrt(((U * (2.0 * n)) * t));
	double tmp;
	if (U <= -2e+16) {
		tmp = t_1;
	} else if (U <= 1.3e+118) {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((U * (2.0 * n)) * t))
	tmp = 0
	if U <= -2e+16:
		tmp = t_1
	elif U <= 1.3e+118:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t))
	tmp = 0.0
	if (U <= -2e+16)
		tmp = t_1;
	elseif (U <= 1.3e+118)
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((U * (2.0 * n)) * t));
	tmp = 0.0;
	if (U <= -2e+16)
		tmp = t_1;
	elseif (U <= 1.3e+118)
		tmp = sqrt((n * (t * (2.0 * U))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -2e+16], t$95$1, If[LessEqual[U, 1.3e+118], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U < -2e16 or 1.30000000000000008e118 < U

   1. Initial program 70.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 58.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -2e16 < U < 1.30000000000000008e118

   1. Initial program 46.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 54.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 29.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 29.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   8. 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. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot U\right) \cdot t\right), n\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), t\right), n\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U \cdot 2\right), t\right), n\right)\right) \]

      7. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, 2\right), t\right), n\right)\right) \]

   9. Applied egg-rr 34.7%

      \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot 2\right) \cdot t\right) \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{if}\;U \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* U (* 2.0 n)) t))))
   (if (<= U -2e+19)
     t_1
     (if (<= U 1.32e+118) (sqrt (* (* 2.0 n) (* U t))) t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((U * (2.0 * n)) * t));
	double tmp;
	if (U <= -2e+19) {
		tmp = t_1;
	} else if (U <= 1.32e+118) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = 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 = sqrt(((u * (2.0d0 * n)) * t))
    if (u <= (-2d+19)) then
        tmp = t_1
    else if (u <= 1.32d+118) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = 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 = Math.sqrt(((U * (2.0 * n)) * t));
	double tmp;
	if (U <= -2e+19) {
		tmp = t_1;
	} else if (U <= 1.32e+118) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt(((U * (2.0 * n)) * t))
	tmp = 0
	if U <= -2e+19:
		tmp = t_1
	elif U <= 1.32e+118:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t))
	tmp = 0.0
	if (U <= -2e+19)
		tmp = t_1;
	elseif (U <= 1.32e+118)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(((U * (2.0 * n)) * t));
	tmp = 0.0;
	if (U <= -2e+19)
		tmp = t_1;
	elseif (U <= 1.32e+118)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -2e+19], t$95$1, If[LessEqual[U, 1.32e+118], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U < -2e19 or 1.3199999999999999e118 < U

   1. Initial program 70.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 58.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -2e19 < U < 1.3199999999999999e118

   1. Initial program 46.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 54.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right), \left(2 \cdot n\right)\right)\right) \]

   6. Applied egg-rr 62.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot t\right)}, \mathsf{*.f64}\left(n, 2\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 34.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]

   9. Simplified 34.7%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(n \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;U \leq 1.32 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.18e-123)
   (sqrt (* (* U (* 2.0 n)) t))
   (pow (* U (* t (* 2.0 n))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.18e-123) {
		tmp = sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = pow((U * (t * (2.0 * n))), 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 <= 1.18d-123) then
        tmp = sqrt(((u * (2.0d0 * n)) * t))
    else
        tmp = (u * (t * (2.0d0 * n))) ** 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 <= 1.18e-123) {
		tmp = Math.sqrt(((U * (2.0 * n)) * t));
	} else {
		tmp = Math.pow((U * (t * (2.0 * n))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.18e-123:
		tmp = math.sqrt(((U * (2.0 * n)) * t))
	else:
		tmp = math.pow((U * (t * (2.0 * n))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.18e-123)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t));
	else
		tmp = Float64(U * Float64(t * Float64(2.0 * n))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.18e-123)
		tmp = sqrt(((U * (2.0 * n)) * t));
	else
		tmp = (U * (t * (2.0 * n))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.18e-123], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Power[N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.18e-123

   1. Initial program 57.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 44.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if 1.18e-123 < l

   1. Initial program 44.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

      3. *-lowering-*.f64 26.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

   7. Simplified 26.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right), \frac{1}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(t \cdot \left(n \cdot 2\right)\right)\right), \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(U \cdot \left(\left(2 \cdot n\right) \cdot t\right)\right), \frac{1}{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \left(\left(2 \cdot n\right) \cdot t\right)\right), \frac{1}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \left(t \cdot \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \left(2 \cdot n\right)\right)\right), \frac{1}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \left(n \cdot 2\right)\right)\right), \frac{1}{2}\right) \]

      13. *-lowering-*.f64 32.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, 2\right)\right)\right), \frac{1}{2}\right) \]

   9. Applied egg-rr 32.0%

      \[\leadsto \color{blue}{{\left(U \cdot \left(t \cdot \left(n \cdot 2\right)\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.18 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* 2.0 n) (* U t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((2.0 * n) * (U * 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 * n) * (u * 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 * n) * (U * t)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((2.0 * n) * (U * t)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * n) * Float64(U * t)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((2.0 * n) * (U * t)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

3. Simplified 61.2%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right) \cdot U\right), \left(2 \cdot n\right)\right)\right) \]

6. Applied egg-rr 62.2%

   \[\leadsto \sqrt{\color{blue}{\left(\left(t + \frac{\frac{U* - U}{\frac{Om}{n \cdot \ell}} + \ell \cdot -2}{\frac{Om}{\ell}}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

7. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(U \cdot t\right)}, \mathsf{*.f64}\left(n, 2\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 38.2%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, t\right), \mathsf{*.f64}\left(n, 2\right)\right)\right) \]

9. Simplified 38.2%

   \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(n \cdot 2\right)} \]

10. Final simplification 38.2%

    \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)} \]
11. Add Preprocessing

Alternative 22: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot n\right) \cdot U\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \left(t + \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(n, U\right)\right), \mathsf{+.f64}\left(t, \left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right)\right)\right) \]

3. Simplified 61.2%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(U* - U\right) \cdot \frac{n \cdot \ell}{Om} + \ell \cdot -2\right)\right)}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right)\right) \]

   3. *-lowering-*.f64 37.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right)\right) \]

7. Simplified 37.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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*))))))